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A classification of edge-colored graphs based on properly colored walks
Discrete Applied Mathematics xxx (xxxx) xxx
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A classification of edge-colored graphs based on properly colored walks ∗
Ruonan Li , Binlong Li, Shenggui Zhang School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, PR China Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, PR China
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Article history: Received 19 September 2019 Received in revised form 7 February 2020 Accepted 7 February 2020 Available online xxxx Keywords: Edge-colored graphs Properly colored Cycles Walks Digraphs
a b s t r a c t A properly colored walk in an edge-colored graph is a walk such that consecutive edges are of distinct colors. In this paper, based on a transformation from directed graphs to edge-colored graphs, we classified edge-colored graphs into three families: degenerate edge-colored graphs, semi-degenerate edge-colored graphs and non-degenerate graphs. By a polynomial-time computable parameter related to properly colored walks, we gave a characterization of these three families. Applying this characterization, we slightly strengthened Yeo’s Theorem (Every edge-colored graph G containing no PC cycle contains a vertex z ∈ V (G) such that each component of G − z is joint to z with at most one color, Yeo, 1997). © 2020 Elsevier B.V. All rights reserved.
1. Introduction Graphs considered in this paper could have multiple edges or loops. For terminology and notation not defined here, we refer the reader to [1]. Let G be an undirected graph. An edge-coloring of G is a mapping col : E(G) → N, where N is the natural number set. A graph G is called an edge-colored graph if G is assigned an edge-coloring. Denote by col(G) the set of colors assigned to E(G). The color degree of a vertex v , denoted by dc (v ), is the number of distinct colors assigned to the edges incident to v . The minimum color degree of G is denoted by δ c (G) = min{dc (v ) : v ∈ V (G)}. An edge-colored graph is called properly colored (or PC ) if each pair of adjacent edges are of distinct colors. Denote by col(e) the color of an edge e. For two vertices u and v , we use col(uv ) to denote the set of colors assigned to the edges joining u and v . When G has no multiple edges, we use col(uv ) to denote the color of edge uv . For a vertex v and a subgraph H of G − v , we use col(v, H) to denote the set of colors assigned to all the edges between v and H. Edge-colored graphs have many special properties. To name a few, Gallai [6] showed that each edge-colored complete graph containing no PC triangle is homomorphic to a 2-colored graph. Yeo [13] proved that an edge-colored graph G containing no PC cycle must contain a vertex z such that each component of G − z is joint to z with at most one color. As a corollary, Wang and Li [12] showed that for each integer k, there exists an edge-colored graph G satisfying δ c (G) = k but containing no PC cycle. Fujita et al. [3] proved that a minimum edge-colored graph G with δ c (G) = 2 is not necessarily a PC cycle, but can also be a generalized bowtie.1 Despite the many special structural properties, some edge-colored graphs show very strong characters of digraphs based on the PC cycle related problems. ∗ Corresponding author at: School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, PR China. E-mail addresses: [email protected] (R. Li), [email protected] (B. Li), [email protected] (S. Zhang). 1 A graph obtained from two vertex-disjoint cycles by joining a path or by identifying one vertex is called a generalized bowtie. https://doi.org/10.1016/j.dam.2020.02.008 0166-218X/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Construction 1. Let G be an edge-colored graph admitting a mapping f : V (G) → N such that col(e) = f (u) or col(e) = f (v ) for each edge e joining u and v . Construct a directed graph D with V (D) = V (G) and uv ∈ A(D) if and only if there exists an edge e joining u and v with col(e) = f (u) and col(e) ̸ = f (v ). In the above construction, since each edge e joining u and v with f (u) = f (v ) is not contained in any PC cycle, we ignore this kind of edges when constructing D. It is easy to check that in Construction 1, each PC cycle in G is also a directed cycle in D and vise versa. So studying PC cycles in this kind of edge-colored graphs is actually a task about cycles in digraphs (see [11] for more details on this opinion). Particularly, when G is a complete graph, the obtained digraph D from Construction 1 is a multipartite tournament. Li et al. [11] called this kind of edge-colored complete graph ‘‘essentially a multipartite tournament’’ (or ‘‘essentially a strongly-connected multipartite tournament’’ when D is strongly connected). Similar constructions are also given in [4] and [5]. To describe such observations, we say those directed graphs analogous edge-colored graphs are degenerate and give the following more general definition. Definition 1. Let G be an edge-colored graph. If there exists a nonempty set S ⊆ V (G) and a function f : S → N such that for each edge e joining u and v , the following holds:
{ col(e) =
f (u) or f (v ),
if u, v ∈ S;
f (u),
if u ∈ S , v ∈ V (G) \ S.
then we say G is semi-degenerate, S is a degenerate set of G and f is compatible to (G, S). In particular, if S = V (G), then we say G is degenerate and say f is compatible to G. If S does not exist, then we say G is non-degenerate. Li et al. [11] tried to describe edge-colored complete graphs which are degenerate and obtained the following. Theorem 1 (Li et al. [11]). Let G be a color degree critical2 edge-colored complete graph. Then G contains no PC theta graph3 if and only if G is degenerate, unless δ c (G) = 2 and G is an edge-colored K4 containing a monochromatic edge-cut. In the other paper by Li et al. [10], they discussed the relation between edge-colored complete graphs and multipartite tournaments on vertex-disjoint cycles with the following result as a core lemma. Theorem 2 (Li et al. [10]). Let G be an edge-colored complete graph with δ c (G) ≥ 2. If v ∈ V (G) is not contained in any PC cycle, then G admits a vertex partition V0 , V1 , V2 , . . . , Vp such that the following statements hold for some distinct colors c1 , c2 , . . . , cp ∈ col(G). (a) 2 ≤ p ≤ dc (v ), v ∈ V0 and |Vi | ≥ 1 for 0 ≤ i ≤ p; (b) col(V0 , Vi ) = {ci } for 1 ≤ i ≤ p; (c) col(Vi , Vj ) ⊆ {ci , cj } for 1 ≤ i < j ≤ p; (d) col(G[Vi ]) ⊆ {ci } (i.e., col(G[Vi ]) = {ci } when |Vi | ≥ 2) for 1 ≤ i ≤ p. We can see that the edge-colored graph G in above theorem is semi-degenerate. Gutin et al. [8] classified edge-colored graphs which containing no PC cycle into five Types. It is easy to check that Types 3,4 and 5 are degenerate. Recently, Čada et al. [2] give a new structural result on edge-colored complete bipartite graphs without containing PC cycle of length 4, which strengthens the result in [4]. The structure is actually degenerate. Theorem 3 (Čada et al. [2]). Let G be an edge-colored complete bipartite graph with δ c (G) ≥ 2 and containing no PC cycle of length 4. Then G admits a function f : V (G) → N such that for each edge xy ∈ E(G), either col(xy) = f (x) or col(xy) = f (y). In this paper, by a method of PC walks, necessary and sufficient conditions are obtained for an edge-colored graph to be degenerate, semi-degenerate and non-degenerate, respectively. We also prove that the three properties are polynomial-time checkable. For other results on PC walks in edge-colored graphs, we refer the reader to [8] and [9]. 2. PC walks A walk W = u0 e0 u1 · · · ek−1 uk (k ≥ 1) is a sequence of alternating vertices and edges such that ui and ui+1 are end vertices of the edge ei for all i ∈ [0, k − 1]. For a walk W = u0 e0 u1 · · · ek−1 uk , we use W −1 to denote the inverse walk of W , namely W −1 = uk ek−1 · · · u1 e0 u0 . Let G be an edge-colored graph and let W = u0 e0 u1 · · · ek−1 uk be a walk in G. If col(ei ) ̸ = col(ei+1 ) for all i with 0 ≤ i ≤ k − 2 or k = 1, then we say W is a properly colored walk (or PC walk). If a PC walk W = u0 e0 u1 · · · ek−1 uk satisfies that u0 = uk and col(e0 ) ̸ = col(ek−1 ), then we say W is a properly colored closed walk (or PC closed walk). 2 An edge-colored graph G is color degree critical if δ c (G − S) < δ c (G) for each nonempty proper subset S ⊂ V (G). 3 A theta graph is obtained by joining two vertices by three internally disjoint paths. Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Observation 1. Let G be a semi-degenerate edge-colored graph, let S be a degenerate set in G and let W be a PC walk in G with starting and ending vertices contained in G − S. Then W contains no vertex from S. Proof. Let f be a function compatible to (G, S). Suppose that W intersects S. Then without loss of generality, assume that W = v e0 u1 e1 u2 · · · ek−1 uk ek w with v, w ∈ V (G) \ S and ui ∈ S for all i ∈ [1, k]. By the definition of ‘‘semi-degenerate’’, we know that col(e0 ) = f (u1 ). This forces col(ei ) = f (ui+1 ) for all i ∈ [1, k − 1]. Note that col(ek ) = f (uk ). We get col(ek−1 ) = col(ek ) = f (uk ), a contradiction. □ Remark 1. Observation 1 implies that each PC closed walk, particularly PC cycle, is either contained in a degenerate set or disjoint with any degenerate set. Observation 2. Let G be an edge-colored graph and let W be a PC walk in G from vi to vj satisfying that the starting color is k, the ending color is ℓ and the length of W is as small as possible. Then for an edge e and its end vertices u and v , the sequence uev appears at most once on W (i.e., each edge appears at most twice and each loop appears at most once on W ). Proof. Suppose that the sequence uev appears at least twice on W . We can assume that W = W1 uev W2 uev W3 . Remove the segment W2 uev from W . Then W1 uev W3 is a shorter PC walk from vi to vj with the starting color k and ending color ℓ. This contradicts to the assumption of W . □ 3. A classification of edge-colored graphs by PC walks Let G be an edge-colored graph and let v be a vertex in G. We say v is covered by a color α if there exists a PC walk W in G from v to itself, say W = v e0 u1 e1 u2 · · · ek−1 uk ek v with k ≥ 0, such that col(e0 ) = col(ek ) = α . We call such a PC walk as a v -covering PC walk of v or a (v, α ) PC walk. In particular, if e is a loop incident to v with col(e) = α , then v ev is a (v, α ) PC walk. Let dw G (v ) denote the number of distinct colors that cover v in G. Before giving the main theorem, we need the following two lemmas which deal with the case that each vertex is covered by at most one color. Let H be a w w subgraph of G. Clearly dw G (v ) ≥ dH (v ) for each v ∈ V (H). When there is no ambiguity, for convenience, we use d (v ) to ( v ). denote dw G Lemma 1. Let G be an edge-colored graph such that dw (u) ≤ 1 for each vertex u ∈ V (G). Let S0 = {u : dw (u) = 0}. If G[S0 ] is degenerate or S0 = ∅, then G is degenerate. Proof. Let S1 = V (G) \ S0 . Then for each vertex u ∈ S1 , there holds dw (u) = 1. Define f (u) to be the unique color which covers u in G. Claim 1. The following statements hold: (a) col(uv ) ⊆ {f (u)} ∪ {f (v )} for distinct vertices u, v ∈ S1 ; (b) col(uw ) ⊆ {f (u)} for u ∈ S1 and w ∈ S0 . Proof. For distinct vertices u, v ∈ S1 , by the definition of f , we know that vertices u and v are covered by colors f (u) and f (v ), respectively. Thus there exist PC walks ue1 Pe′1 u and v e2 Qe′2 v such that col(e1 ) = col(e′1 ) = f (u) and col(e2 ) = col(e′2 ) = f (v ). Suppose that there is an edge e joining u and v such that col(e) ̸ = f (u) and col(e) ̸ = f (v ). Then it is easy to check that uev e2 Qe′2 v eu is a PC walk from u to u, which implies that u is also covered by the color col(e) which is distinct from f (u). Thus dw (u) ≥ 2, a contradiction. So we have col(uv ) ⊆ {f (u)} ∪ {f (v )}. For u ∈ S1 and w ∈ S0 , suppose that there exists an edge e joining u and w such that col(e) ̸ = f (u). Then w eue1 Pe′1 uew is a w -covering PC walk, which contradicts that w ∈ S0 . Hence we have col(uw ) ⊆ {f (u)}. □ If S0 = ∅, then by Claim 1, G is degenerate. Otherwise, since G[S0 ] is degenerate, there exists a function g : V (G[S0 ]) → N compatible to G[S0 ]. Extend function g to S1 by defining g(u) = f (u) for each vertex u ∈ S1 . Then g is compatible to G and therefore G is degenerate. □ Lemma 2.
Let G be an edge-colored graph with dw (u) = 0 for each vertex u ∈ V (G). Then G is degenerate.
Proof. By induction on the order of G. When |G| = 1, let u be the only vertex in G. Since dw (u) = 0, no loop is incident to u. So there is nothing to prove. Now suppose that |G| = n, E(G) ̸ = ∅ and Lemma 2 holds for all edge-colored graphs of order k ∈ [1, n − 1]. Let e0 be an edge in G joining u and v with col(e0 ) = α . Construct a new edge-colored graph Huα,v from G by adding a loop e to u with col(e) = α . Claim 2.
Each vertex in Huα,v is covered by at most one color.
Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Proof. Let H = Huα,v . For a vertex w ∈ V (H)\{u}, if w is covered by 2 distinct colors i and j in H, then any (w, i) PC walk and (w, j) PC walk must contain the adding loop e. By Observation 2, we may assume that the two w -covering PC walks are w hi Wgi ueu · · · h′i w and w hj Qgj ueu · · · h′j w such that w hi Wgi u and w hj Qgj u are contained in G. Note that col(e) = α . We have col(gi ) ̸ = α and col(gj ) ̸ = α . Thus v e0 ugi W −1 hi w hj Qgj ue0 v is a v -covering PC walk in G, a contradiction. So each vertex in V (H)\{u} is covered by at most one color in H. Note that u is covered by the color α in H (consider the loop ueu). Suppose that u is also covered by a color β ̸ = α in H and uhWgueu · · · h′ u is a (u, β ) PC walk with uhWgu contained in G. Then v e0 uhWgue0 v is a v -covering PC walk in G, a contradiction. So u is only covered by the color α in H. In summary, each vertex in Huα,v is covered by at most one color. □ Let S0 be the set of vertices in Huα,v that are not covered by any color. Since u is covered by α , we get |S0 | < n. By Claim 2, Lemma 1 and the induction hypothesis, the edge-colored graph Huα,v is degenerate. Since G is a subgraph of Huα,v , we can see that G is also degenerate. □ We now have all the ingredients for the statements and the proof of the following main theorem. Theorem 4. Let G be an edge-colored graph. Let Si = {u : dw (u) = i} for i = 0, 1 and S = S0 ∪ S1 . Then (a) G is degenerate if and only if S = V (G); (b) G is non-degenerate if and only if S = ∅; (c) G is semi-degenerate if and only if S ̸ = ∅; (d) S is a degenerate set of G unless S = ∅; (e) S1 is a degenerate set of G unless S1 = ∅; (f ) there is no edge between S0 and G − S; (g) every degenerate set of G is a subset of S; (h) if G is non-degenerate, then dw (v ) = dc (v ) for every vertex v ∈ V (G). Proof. We first prove Theorem 4 (a). Lemmas 1 and 2 imply the sufficiency immediately. For the necessity, if G is degenerate, then there exists a function f which is compatible to G. Assume that v is a vertex in G covered by a color α . Then G contains a (v, α ) PC walk v e0 u1 e1 u2 · · · ek−1 uk ek v (k ≥ 0). We assert that f (v ) = α . Suppose to the contrary that f (v ) ̸ = α . Then we have f (u1 ) = α and f (uj ) = col(ej−1 ) for all j with 2 ≤ j ≤ k. In particular, f (uk ) ̸ = col(ek ). Note that col(ek ) = α and f (v ) ̸ = α . This contradicts that f is compatible to G. So f (v ) = α . If v is also covered by a color β , then similarly we can obtain f (v ) = β . Thus there must hold α = β . Hence each vertex in G can be covered by at most one color. So Theorem 4 (a) is proved. For vertices u ∈ S0 and v ∈ V (G) \ S, if there is an edge e joining u and v , then choose a color, say α , which covers v but distinct to col(e) (this is possible since dw (v ) ≥ 2) and consider a (v, α ) PC walk. We can see that u is covered by col(e), which contradicts that u ∈ S0 . So Theorem 4 (f ) holds. By Lemma 2, G[S0 ] is degenerate. Let f be a compatible function to G[S0 ]. Extend f to each vertex u ∈ S1 by defining f (u) to be the unique color that covers u in G. Then similar to the proof of Claim 1, we can get:
• col(uv ) ⊆ {f (u)} ∪ {f (v )} for distinct vertices u, v ∈ S1 ; • col(uw) ⊆ {f (u)} for u ∈ S1 and w ∈ S0 ; • col(uw) ⊆ {f (u)} for u ∈ S1 and w ∈ V (G) \ S (otherwise we will get dw (u) ≥ 2). Hence f is compatible to (G, S1 ) when S1 ̸ = ∅. Recall Theorem 4 (f ). We can see that f is also compatible to (G, S) when S ̸ = ∅. Therefore Theorem 4 (d) and (e) hold. Now we will prove Theorem 4 (g). For a vertex u ∈ V (G) \ S, since u is covered by at least 2 colors, we can find a (u, α ) PC walk and a (u, β ) PC walk with α ̸ = β . By combining these two PC walks together, we get a PC closed walk W passing through u. Observation 1 implies that each PC closed walk is either contained in a degenerate set or disjoint with any degenerate set. Suppose that u is contained in a degenerate set S ′ . Then W is contained in G[S ′ ] and u is covered by α and β in G[S ′ ]. However, note that G[S ′ ] is degenerate. By Theorem 4 (a), each vertex in G[S ′ ] is covered by at most 1 color, a contradiction. So each vertex in V (G) \ S is not contained in any degenerate set, namely Theorem 4 (g) holds. By Theorem 4 (d) and (g), we can easily get Theorem 4 (c), which immediately implies Theorem 4 (b). Theorem 4 (h) can be obtained directly from Theorem 4 (b) as following: for each vertex v and each edge e incident to v , let u be the other end of e. Since G is non-degenerate, by Theorem 4 (b), dw (u) ≥ 2. So we can find a (u, α ) PC walk with α ̸ = col(e). By adding e to this (u, α ) PC walk, we obtain a (v, col(e)) PC walk, which implies that col(e) covers v . Therefore we have dw (v ) = dc (v ) when G is non-degenerate. The proof is complete. □ Let us say v is of type I if dw (v ) = 0, of type II if dw (v ) = 1 and of type III if dw (v ) ≥ 2. To decide an edge-colored graph is degenerate, non-degenerate or semi-degenerate, by Theorem 4, it suffices to decide the type of v for each vertex v ∈ V (G). Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Theorem 5. Let G be an edge-colored graph. It is polynomial-time checkable whether G is degenerate, non-degenerate or semi-degenerate. Proof. Without loss of generality, we may assume that |V (G)| = n and
• Two edges joining a same pair of vertices always have distinct colors; • For each vertex v ∈ V (G), there are at most 2 differently colored loops incident to v ; • Each pair of vertices u and v are joint by at most 3 differently colored edges; (n) Thus |col(G)| ≤ |E(G)| = m ≤ 3 2 + 2n. It is easy to see that more colored edges or loops will not change the type of any
vertex. (t) Assume V (G) = {v1 , v2 , . . . , vn }. Let ai,j,k.ℓ be the number of PC walks of length t from vi to vj such that the starting color is k and ending color is ℓ. Then (1) ai,j,k,l
{ =
1,
if vi ∈ N(vj ), k = ℓ and k ∈ col(vi vj );
0,
otherwise.
Note that if vi is incident to a loop, then vi ∈ N(vi ). For t ≥ 2, we have (t)
ai,j,k,ℓ =
∑
∑
(t −1)
(1)
ai,s,k,p as,j,q,ℓ .
(1)
p̸ =q 1≤s≤n p,q∈col(G)
By Observation 2, if there exists a PC walk from vi to vj such that the starting color is k and ending color is ℓ, then there ∑ (t) (t) exists an integer t ∈ [1, 2m] such that ai,j,k,ℓ ≥ 1. Let bi,k = 1≤t ≤2m ai,i,k,k . Then bi,k is positive if and only if the color k covers vi . Hence dw (vi ) = |{k : k ∈ col(G), bi,k > 0}|. To decide the computational complexity of dw (vi ), we need more notation. For each positive integer t, let A(t) be a (t) 4-dimension matrix with A(t) = (ai,j,k,l )n×n×c ×c . Here c = |col(G)| = O(n2 ). By Eq. (1), calculating A(t) when A(1) and (t −1) 3 4 A are known needs O(n c ) = O(n11 ) time. Hence we can get A(t) from G in O(tn11 ) time, which implies that bi,k can be obtained in O(m2 n11 ) = O(n15 ) time and therefore dw (vi ) can be calculated in O(cn15 ) = O(n17 ) time. By Theorem 4, whether G is degenerate/semi-degenerate/non-degenerate is polynomial-time checkable. □ 4. PC walks and Yeo’s theorem In the theory of PC cycles, one of the most important theorems should be the following local characterization given by Yeo4 [13] Theorem 6 (Yeo [13]). Let G be an edge-colored graph containing no PC cycle. Then there exists a vertex z ∈ V (G) such that no component of G − z is joined to z with edges of more than one color. By using the notation of PC walks, now we strengthen the statement of the above theorem. We say a walk W from u to v is a reaching walk if v only appears at the end of W . Define Svi = {u : there exists a reaching PC walk from u to v with ending color i}. Theorem 7. Let G be an edge-colored graph containing no PC cycle. Then one of the following holds. (1) G is semi-degenerate with δ c (G) ≤ 1; (2) G is non-degenerate and there exists a vertex z such that for each component F of G − z, F is joint to z by at most one color. Moreover, if col(z , F ) = {i}, then for each edge e in F , there exists a (z , i) PC walk contained in F + z and passing through e. Proof of Theorem 7. Without loss of generality, assume that G is connected with |G| ≥ 2 and contains no loops or multiple edges (since a loop does not effect the existence of PC cycles and two differently colored edges joining a same pair of vertices form a PC cycle of length 2, which contradicts that G is PC acyclic). If G is semi-degenerate, then let S be a degenerate set of G and let f be a function compatible to (G, S). Construct a digraph H with V (H) = S and A(H) = {uv : col(uv ) = f (u) and col(uv ) ̸ = f (v )}. 4 In 1983, Grossman and Häggkvist [7] firstly obtained the structure for 2-colored graphs. In 1997, Yeo [13] extended the result to edge-colored graphs of arbitrary number of colors. Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Since G contains no PC cycle, H has no directed cycle. Thus there exists a vertex v ∈ S such that d− H (v ) = 0. By the construction of H and the definition of degenerate set, we can see that col(v, G − v ) = {f (v )}. Thus δ c (G) ≤ 1. Now assume that G is non-degenerate. Then by Theorem 4 (a), dw (u) ≥ 2 for each u ∈ V (G). By Theorem 6, let z be a vertex in G such that no component of G − z is joined to z with edges of more than one color. Let F be a component of G − z with col(z , F ) = {i}. First we prove that each vertex in F has a reaching PC walk to z. It suffices to show that if u ∈ V (F ) ∩ Szi , then each w ∈ NF (u) is contained in Szi . Let ueWhz be a reaching PC walk from u to z with ending color i. Let e′ be an edge joining w and u. If w is contained in W or col(e′ ) ̸ = col(e), then there is nothing to prove. So we may assume that w is not contained in W and col(e′ ) = col(e). Since dw (u) ≥ 2, G contains a (u, α ) PC walk ue1 W ′ e2 u such that α ̸ = col(e). If z is not contained in W ′ , then w e′ ue1 W ′ e2 ueWhz is a reaching PC walk from w to z with ending color i. Otherwise, we can find a reaching PC walk in W ′ from u to z such that the starting color is col(e1 ) and ending color is i (since col(z , F ) = {i}). Together with the edge e′ , we can see that w ∈ Szi . Now let e be an arbitrary edge in F joining u and w . We assert that there exist two reaching PC walks: W from u to z and W ′ from w to z such that neither of the starting colors is col(e). Because, from above, we can always obtain two reaching PC walks to z from u and w . Then consider a (u, α ) PC walk with α ̸ = col(e) and a (w, β ) PC walk with β ̸ = col(e), and revising the two reaching PC walks to z from u and w when it is necessary. The proof is complete. □ Remark 2. By Observation 1, more efforts should be paid to non-degenerate edge-colored graphs when studying PC cycle related problems. For instance, the minimum color degree condition given by Fujita and Magnant [5] for the existence of PC Hamilton cycles is true when the edge-colored complete graph is semi-degenerate and might be improved in the non-degenerate case. CRediT authorship contribution statement Ruonan Li: Conceptualization, Methodology, Writing - original draft. Binlong Li: Methodology. Shenggui Zhang: Writing - review & editing. Acknowledgments The authors are grateful to the referees for their detailed comments and valuable suggestions that greatly improved this article. The research is supported by NSFC (Nos. 11601429, 11671320 and 11901459), and the Fundamental Research Funds for the Central Universities of China (Nos. 31020180QD124 and 3102019GHJD003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.A. Bondy, U.S.R. Murty, Graph Theory, in: Springer Graduate Texts in Mathematics, vol. 244, 2008. R. Čada, K. Ozeki, K. Yoshimoto, A complete bipartite graph without properly colored cycles of length four, J. Graph Theory 93 (2020) 168–180. S. Fujita, R. Li, G. Wang, Decomposing edge-colored graphs under color degree constraints, Combin. Probab. Comput. 28 (2019) 755–767. S. Fujita, R. Li, S. Zhang, Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs, J. Graph Theory 87 (2018) 362–373. S. Fujita, C. Magnant, Properly colored paths and cycles, Discrete Appl. Math. 159 (2011) 1391–1397. T. Gallai, Transitiv orientierbare graphen, Acta Math. Acad. Sci. Hungar. 18 (1967) 25–66. J.W. Grossman, R. Häggkvist, Alternating cycles in edge-partitioned graphs, J. Combin. Theory Ser. B 34 (1983) 77–81. G. Gutin, M. Jones, B. Sheng, M. Wahlström, A. Yeo, Acyclicity in edge-colored graphs, Discrete Math. 340 (2017) 1–8. G. Gutin, M. Jones, B. Sheng, M. Wahlström, A. Yeo, Chinese postman problem on edge-colored multigraphs, Discrete Appl. Math. 217 (2017) 196–202. R. Li, H. Broersma, S. Zhang, Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs, J. Graph Theory, https://doi.org/10. 1002/jgt.22536. R. Li, H. Broersma, S. Zhang, Properly edge-colored theta graphs in edge-colored complete graphs, Graphs Combin. 35 (2019) 261–286. G. Wang, H. Li, Color degree and alternating cycles in edge-colored graphs, Discrete Math. 309 (2009) 4349–4354. A. Yeo, A note on alternating cycles in edge-colored graphs, J. Combin. Theory Ser. B 69 (1997) 222–225.
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A classification of edge-colored graphs based on properly colored walks ∗
Ruonan Li , Binlong Li, Shenggui Zhang School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, PR China Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, PR China
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Article history: Received 19 September 2019 Received in revised form 7 February 2020 Accepted 7 February 2020 Available online xxxx Keywords: Edge-colored graphs Properly colored Cycles Walks Digraphs
a b s t r a c t A properly colored walk in an edge-colored graph is a walk such that consecutive edges are of distinct colors. In this paper, based on a transformation from directed graphs to edge-colored graphs, we classified edge-colored graphs into three families: degenerate edge-colored graphs, semi-degenerate edge-colored graphs and non-degenerate graphs. By a polynomial-time computable parameter related to properly colored walks, we gave a characterization of these three families. Applying this characterization, we slightly strengthened Yeo’s Theorem (Every edge-colored graph G containing no PC cycle contains a vertex z ∈ V (G) such that each component of G − z is joint to z with at most one color, Yeo, 1997). © 2020 Elsevier B.V. All rights reserved.
1. Introduction Graphs considered in this paper could have multiple edges or loops. For terminology and notation not defined here, we refer the reader to [1]. Let G be an undirected graph. An edge-coloring of G is a mapping col : E(G) → N, where N is the natural number set. A graph G is called an edge-colored graph if G is assigned an edge-coloring. Denote by col(G) the set of colors assigned to E(G). The color degree of a vertex v , denoted by dc (v ), is the number of distinct colors assigned to the edges incident to v . The minimum color degree of G is denoted by δ c (G) = min{dc (v ) : v ∈ V (G)}. An edge-colored graph is called properly colored (or PC ) if each pair of adjacent edges are of distinct colors. Denote by col(e) the color of an edge e. For two vertices u and v , we use col(uv ) to denote the set of colors assigned to the edges joining u and v . When G has no multiple edges, we use col(uv ) to denote the color of edge uv . For a vertex v and a subgraph H of G − v , we use col(v, H) to denote the set of colors assigned to all the edges between v and H. Edge-colored graphs have many special properties. To name a few, Gallai [6] showed that each edge-colored complete graph containing no PC triangle is homomorphic to a 2-colored graph. Yeo [13] proved that an edge-colored graph G containing no PC cycle must contain a vertex z such that each component of G − z is joint to z with at most one color. As a corollary, Wang and Li [12] showed that for each integer k, there exists an edge-colored graph G satisfying δ c (G) = k but containing no PC cycle. Fujita et al. [3] proved that a minimum edge-colored graph G with δ c (G) = 2 is not necessarily a PC cycle, but can also be a generalized bowtie.1 Despite the many special structural properties, some edge-colored graphs show very strong characters of digraphs based on the PC cycle related problems. ∗ Corresponding author at: School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, PR China. E-mail addresses: [email protected] (R. Li), [email protected] (B. Li), [email protected] (S. Zhang). 1 A graph obtained from two vertex-disjoint cycles by joining a path or by identifying one vertex is called a generalized bowtie. https://doi.org/10.1016/j.dam.2020.02.008 0166-218X/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Construction 1. Let G be an edge-colored graph admitting a mapping f : V (G) → N such that col(e) = f (u) or col(e) = f (v ) for each edge e joining u and v . Construct a directed graph D with V (D) = V (G) and uv ∈ A(D) if and only if there exists an edge e joining u and v with col(e) = f (u) and col(e) ̸ = f (v ). In the above construction, since each edge e joining u and v with f (u) = f (v ) is not contained in any PC cycle, we ignore this kind of edges when constructing D. It is easy to check that in Construction 1, each PC cycle in G is also a directed cycle in D and vise versa. So studying PC cycles in this kind of edge-colored graphs is actually a task about cycles in digraphs (see [11] for more details on this opinion). Particularly, when G is a complete graph, the obtained digraph D from Construction 1 is a multipartite tournament. Li et al. [11] called this kind of edge-colored complete graph ‘‘essentially a multipartite tournament’’ (or ‘‘essentially a strongly-connected multipartite tournament’’ when D is strongly connected). Similar constructions are also given in [4] and [5]. To describe such observations, we say those directed graphs analogous edge-colored graphs are degenerate and give the following more general definition. Definition 1. Let G be an edge-colored graph. If there exists a nonempty set S ⊆ V (G) and a function f : S → N such that for each edge e joining u and v , the following holds:
{ col(e) =
f (u) or f (v ),
if u, v ∈ S;
f (u),
if u ∈ S , v ∈ V (G) \ S.
then we say G is semi-degenerate, S is a degenerate set of G and f is compatible to (G, S). In particular, if S = V (G), then we say G is degenerate and say f is compatible to G. If S does not exist, then we say G is non-degenerate. Li et al. [11] tried to describe edge-colored complete graphs which are degenerate and obtained the following. Theorem 1 (Li et al. [11]). Let G be a color degree critical2 edge-colored complete graph. Then G contains no PC theta graph3 if and only if G is degenerate, unless δ c (G) = 2 and G is an edge-colored K4 containing a monochromatic edge-cut. In the other paper by Li et al. [10], they discussed the relation between edge-colored complete graphs and multipartite tournaments on vertex-disjoint cycles with the following result as a core lemma. Theorem 2 (Li et al. [10]). Let G be an edge-colored complete graph with δ c (G) ≥ 2. If v ∈ V (G) is not contained in any PC cycle, then G admits a vertex partition V0 , V1 , V2 , . . . , Vp such that the following statements hold for some distinct colors c1 , c2 , . . . , cp ∈ col(G). (a) 2 ≤ p ≤ dc (v ), v ∈ V0 and |Vi | ≥ 1 for 0 ≤ i ≤ p; (b) col(V0 , Vi ) = {ci } for 1 ≤ i ≤ p; (c) col(Vi , Vj ) ⊆ {ci , cj } for 1 ≤ i < j ≤ p; (d) col(G[Vi ]) ⊆ {ci } (i.e., col(G[Vi ]) = {ci } when |Vi | ≥ 2) for 1 ≤ i ≤ p. We can see that the edge-colored graph G in above theorem is semi-degenerate. Gutin et al. [8] classified edge-colored graphs which containing no PC cycle into five Types. It is easy to check that Types 3,4 and 5 are degenerate. Recently, Čada et al. [2] give a new structural result on edge-colored complete bipartite graphs without containing PC cycle of length 4, which strengthens the result in [4]. The structure is actually degenerate. Theorem 3 (Čada et al. [2]). Let G be an edge-colored complete bipartite graph with δ c (G) ≥ 2 and containing no PC cycle of length 4. Then G admits a function f : V (G) → N such that for each edge xy ∈ E(G), either col(xy) = f (x) or col(xy) = f (y). In this paper, by a method of PC walks, necessary and sufficient conditions are obtained for an edge-colored graph to be degenerate, semi-degenerate and non-degenerate, respectively. We also prove that the three properties are polynomial-time checkable. For other results on PC walks in edge-colored graphs, we refer the reader to [8] and [9]. 2. PC walks A walk W = u0 e0 u1 · · · ek−1 uk (k ≥ 1) is a sequence of alternating vertices and edges such that ui and ui+1 are end vertices of the edge ei for all i ∈ [0, k − 1]. For a walk W = u0 e0 u1 · · · ek−1 uk , we use W −1 to denote the inverse walk of W , namely W −1 = uk ek−1 · · · u1 e0 u0 . Let G be an edge-colored graph and let W = u0 e0 u1 · · · ek−1 uk be a walk in G. If col(ei ) ̸ = col(ei+1 ) for all i with 0 ≤ i ≤ k − 2 or k = 1, then we say W is a properly colored walk (or PC walk). If a PC walk W = u0 e0 u1 · · · ek−1 uk satisfies that u0 = uk and col(e0 ) ̸ = col(ek−1 ), then we say W is a properly colored closed walk (or PC closed walk). 2 An edge-colored graph G is color degree critical if δ c (G − S) < δ c (G) for each nonempty proper subset S ⊂ V (G). 3 A theta graph is obtained by joining two vertices by three internally disjoint paths. Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Observation 1. Let G be a semi-degenerate edge-colored graph, let S be a degenerate set in G and let W be a PC walk in G with starting and ending vertices contained in G − S. Then W contains no vertex from S. Proof. Let f be a function compatible to (G, S). Suppose that W intersects S. Then without loss of generality, assume that W = v e0 u1 e1 u2 · · · ek−1 uk ek w with v, w ∈ V (G) \ S and ui ∈ S for all i ∈ [1, k]. By the definition of ‘‘semi-degenerate’’, we know that col(e0 ) = f (u1 ). This forces col(ei ) = f (ui+1 ) for all i ∈ [1, k − 1]. Note that col(ek ) = f (uk ). We get col(ek−1 ) = col(ek ) = f (uk ), a contradiction. □ Remark 1. Observation 1 implies that each PC closed walk, particularly PC cycle, is either contained in a degenerate set or disjoint with any degenerate set. Observation 2. Let G be an edge-colored graph and let W be a PC walk in G from vi to vj satisfying that the starting color is k, the ending color is ℓ and the length of W is as small as possible. Then for an edge e and its end vertices u and v , the sequence uev appears at most once on W (i.e., each edge appears at most twice and each loop appears at most once on W ). Proof. Suppose that the sequence uev appears at least twice on W . We can assume that W = W1 uev W2 uev W3 . Remove the segment W2 uev from W . Then W1 uev W3 is a shorter PC walk from vi to vj with the starting color k and ending color ℓ. This contradicts to the assumption of W . □ 3. A classification of edge-colored graphs by PC walks Let G be an edge-colored graph and let v be a vertex in G. We say v is covered by a color α if there exists a PC walk W in G from v to itself, say W = v e0 u1 e1 u2 · · · ek−1 uk ek v with k ≥ 0, such that col(e0 ) = col(ek ) = α . We call such a PC walk as a v -covering PC walk of v or a (v, α ) PC walk. In particular, if e is a loop incident to v with col(e) = α , then v ev is a (v, α ) PC walk. Let dw G (v ) denote the number of distinct colors that cover v in G. Before giving the main theorem, we need the following two lemmas which deal with the case that each vertex is covered by at most one color. Let H be a w w subgraph of G. Clearly dw G (v ) ≥ dH (v ) for each v ∈ V (H). When there is no ambiguity, for convenience, we use d (v ) to ( v ). denote dw G Lemma 1. Let G be an edge-colored graph such that dw (u) ≤ 1 for each vertex u ∈ V (G). Let S0 = {u : dw (u) = 0}. If G[S0 ] is degenerate or S0 = ∅, then G is degenerate. Proof. Let S1 = V (G) \ S0 . Then for each vertex u ∈ S1 , there holds dw (u) = 1. Define f (u) to be the unique color which covers u in G. Claim 1. The following statements hold: (a) col(uv ) ⊆ {f (u)} ∪ {f (v )} for distinct vertices u, v ∈ S1 ; (b) col(uw ) ⊆ {f (u)} for u ∈ S1 and w ∈ S0 . Proof. For distinct vertices u, v ∈ S1 , by the definition of f , we know that vertices u and v are covered by colors f (u) and f (v ), respectively. Thus there exist PC walks ue1 Pe′1 u and v e2 Qe′2 v such that col(e1 ) = col(e′1 ) = f (u) and col(e2 ) = col(e′2 ) = f (v ). Suppose that there is an edge e joining u and v such that col(e) ̸ = f (u) and col(e) ̸ = f (v ). Then it is easy to check that uev e2 Qe′2 v eu is a PC walk from u to u, which implies that u is also covered by the color col(e) which is distinct from f (u). Thus dw (u) ≥ 2, a contradiction. So we have col(uv ) ⊆ {f (u)} ∪ {f (v )}. For u ∈ S1 and w ∈ S0 , suppose that there exists an edge e joining u and w such that col(e) ̸ = f (u). Then w eue1 Pe′1 uew is a w -covering PC walk, which contradicts that w ∈ S0 . Hence we have col(uw ) ⊆ {f (u)}. □ If S0 = ∅, then by Claim 1, G is degenerate. Otherwise, since G[S0 ] is degenerate, there exists a function g : V (G[S0 ]) → N compatible to G[S0 ]. Extend function g to S1 by defining g(u) = f (u) for each vertex u ∈ S1 . Then g is compatible to G and therefore G is degenerate. □ Lemma 2.
Let G be an edge-colored graph with dw (u) = 0 for each vertex u ∈ V (G). Then G is degenerate.
Proof. By induction on the order of G. When |G| = 1, let u be the only vertex in G. Since dw (u) = 0, no loop is incident to u. So there is nothing to prove. Now suppose that |G| = n, E(G) ̸ = ∅ and Lemma 2 holds for all edge-colored graphs of order k ∈ [1, n − 1]. Let e0 be an edge in G joining u and v with col(e0 ) = α . Construct a new edge-colored graph Huα,v from G by adding a loop e to u with col(e) = α . Claim 2.
Each vertex in Huα,v is covered by at most one color.
Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Proof. Let H = Huα,v . For a vertex w ∈ V (H)\{u}, if w is covered by 2 distinct colors i and j in H, then any (w, i) PC walk and (w, j) PC walk must contain the adding loop e. By Observation 2, we may assume that the two w -covering PC walks are w hi Wgi ueu · · · h′i w and w hj Qgj ueu · · · h′j w such that w hi Wgi u and w hj Qgj u are contained in G. Note that col(e) = α . We have col(gi ) ̸ = α and col(gj ) ̸ = α . Thus v e0 ugi W −1 hi w hj Qgj ue0 v is a v -covering PC walk in G, a contradiction. So each vertex in V (H)\{u} is covered by at most one color in H. Note that u is covered by the color α in H (consider the loop ueu). Suppose that u is also covered by a color β ̸ = α in H and uhWgueu · · · h′ u is a (u, β ) PC walk with uhWgu contained in G. Then v e0 uhWgue0 v is a v -covering PC walk in G, a contradiction. So u is only covered by the color α in H. In summary, each vertex in Huα,v is covered by at most one color. □ Let S0 be the set of vertices in Huα,v that are not covered by any color. Since u is covered by α , we get |S0 | < n. By Claim 2, Lemma 1 and the induction hypothesis, the edge-colored graph Huα,v is degenerate. Since G is a subgraph of Huα,v , we can see that G is also degenerate. □ We now have all the ingredients for the statements and the proof of the following main theorem. Theorem 4. Let G be an edge-colored graph. Let Si = {u : dw (u) = i} for i = 0, 1 and S = S0 ∪ S1 . Then (a) G is degenerate if and only if S = V (G); (b) G is non-degenerate if and only if S = ∅; (c) G is semi-degenerate if and only if S ̸ = ∅; (d) S is a degenerate set of G unless S = ∅; (e) S1 is a degenerate set of G unless S1 = ∅; (f ) there is no edge between S0 and G − S; (g) every degenerate set of G is a subset of S; (h) if G is non-degenerate, then dw (v ) = dc (v ) for every vertex v ∈ V (G). Proof. We first prove Theorem 4 (a). Lemmas 1 and 2 imply the sufficiency immediately. For the necessity, if G is degenerate, then there exists a function f which is compatible to G. Assume that v is a vertex in G covered by a color α . Then G contains a (v, α ) PC walk v e0 u1 e1 u2 · · · ek−1 uk ek v (k ≥ 0). We assert that f (v ) = α . Suppose to the contrary that f (v ) ̸ = α . Then we have f (u1 ) = α and f (uj ) = col(ej−1 ) for all j with 2 ≤ j ≤ k. In particular, f (uk ) ̸ = col(ek ). Note that col(ek ) = α and f (v ) ̸ = α . This contradicts that f is compatible to G. So f (v ) = α . If v is also covered by a color β , then similarly we can obtain f (v ) = β . Thus there must hold α = β . Hence each vertex in G can be covered by at most one color. So Theorem 4 (a) is proved. For vertices u ∈ S0 and v ∈ V (G) \ S, if there is an edge e joining u and v , then choose a color, say α , which covers v but distinct to col(e) (this is possible since dw (v ) ≥ 2) and consider a (v, α ) PC walk. We can see that u is covered by col(e), which contradicts that u ∈ S0 . So Theorem 4 (f ) holds. By Lemma 2, G[S0 ] is degenerate. Let f be a compatible function to G[S0 ]. Extend f to each vertex u ∈ S1 by defining f (u) to be the unique color that covers u in G. Then similar to the proof of Claim 1, we can get:
• col(uv ) ⊆ {f (u)} ∪ {f (v )} for distinct vertices u, v ∈ S1 ; • col(uw) ⊆ {f (u)} for u ∈ S1 and w ∈ S0 ; • col(uw) ⊆ {f (u)} for u ∈ S1 and w ∈ V (G) \ S (otherwise we will get dw (u) ≥ 2). Hence f is compatible to (G, S1 ) when S1 ̸ = ∅. Recall Theorem 4 (f ). We can see that f is also compatible to (G, S) when S ̸ = ∅. Therefore Theorem 4 (d) and (e) hold. Now we will prove Theorem 4 (g). For a vertex u ∈ V (G) \ S, since u is covered by at least 2 colors, we can find a (u, α ) PC walk and a (u, β ) PC walk with α ̸ = β . By combining these two PC walks together, we get a PC closed walk W passing through u. Observation 1 implies that each PC closed walk is either contained in a degenerate set or disjoint with any degenerate set. Suppose that u is contained in a degenerate set S ′ . Then W is contained in G[S ′ ] and u is covered by α and β in G[S ′ ]. However, note that G[S ′ ] is degenerate. By Theorem 4 (a), each vertex in G[S ′ ] is covered by at most 1 color, a contradiction. So each vertex in V (G) \ S is not contained in any degenerate set, namely Theorem 4 (g) holds. By Theorem 4 (d) and (g), we can easily get Theorem 4 (c), which immediately implies Theorem 4 (b). Theorem 4 (h) can be obtained directly from Theorem 4 (b) as following: for each vertex v and each edge e incident to v , let u be the other end of e. Since G is non-degenerate, by Theorem 4 (b), dw (u) ≥ 2. So we can find a (u, α ) PC walk with α ̸ = col(e). By adding e to this (u, α ) PC walk, we obtain a (v, col(e)) PC walk, which implies that col(e) covers v . Therefore we have dw (v ) = dc (v ) when G is non-degenerate. The proof is complete. □ Let us say v is of type I if dw (v ) = 0, of type II if dw (v ) = 1 and of type III if dw (v ) ≥ 2. To decide an edge-colored graph is degenerate, non-degenerate or semi-degenerate, by Theorem 4, it suffices to decide the type of v for each vertex v ∈ V (G). Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Theorem 5. Let G be an edge-colored graph. It is polynomial-time checkable whether G is degenerate, non-degenerate or semi-degenerate. Proof. Without loss of generality, we may assume that |V (G)| = n and
• Two edges joining a same pair of vertices always have distinct colors; • For each vertex v ∈ V (G), there are at most 2 differently colored loops incident to v ; • Each pair of vertices u and v are joint by at most 3 differently colored edges; (n) Thus |col(G)| ≤ |E(G)| = m ≤ 3 2 + 2n. It is easy to see that more colored edges or loops will not change the type of any
vertex. (t) Assume V (G) = {v1 , v2 , . . . , vn }. Let ai,j,k.ℓ be the number of PC walks of length t from vi to vj such that the starting color is k and ending color is ℓ. Then (1) ai,j,k,l
{ =
1,
if vi ∈ N(vj ), k = ℓ and k ∈ col(vi vj );
0,
otherwise.
Note that if vi is incident to a loop, then vi ∈ N(vi ). For t ≥ 2, we have (t)
ai,j,k,ℓ =
∑
∑
(t −1)
(1)
ai,s,k,p as,j,q,ℓ .
(1)
p̸ =q 1≤s≤n p,q∈col(G)
By Observation 2, if there exists a PC walk from vi to vj such that the starting color is k and ending color is ℓ, then there ∑ (t) (t) exists an integer t ∈ [1, 2m] such that ai,j,k,ℓ ≥ 1. Let bi,k = 1≤t ≤2m ai,i,k,k . Then bi,k is positive if and only if the color k covers vi . Hence dw (vi ) = |{k : k ∈ col(G), bi,k > 0}|. To decide the computational complexity of dw (vi ), we need more notation. For each positive integer t, let A(t) be a (t) 4-dimension matrix with A(t) = (ai,j,k,l )n×n×c ×c . Here c = |col(G)| = O(n2 ). By Eq. (1), calculating A(t) when A(1) and (t −1) 3 4 A are known needs O(n c ) = O(n11 ) time. Hence we can get A(t) from G in O(tn11 ) time, which implies that bi,k can be obtained in O(m2 n11 ) = O(n15 ) time and therefore dw (vi ) can be calculated in O(cn15 ) = O(n17 ) time. By Theorem 4, whether G is degenerate/semi-degenerate/non-degenerate is polynomial-time checkable. □ 4. PC walks and Yeo’s theorem In the theory of PC cycles, one of the most important theorems should be the following local characterization given by Yeo4 [13] Theorem 6 (Yeo [13]). Let G be an edge-colored graph containing no PC cycle. Then there exists a vertex z ∈ V (G) such that no component of G − z is joined to z with edges of more than one color. By using the notation of PC walks, now we strengthen the statement of the above theorem. We say a walk W from u to v is a reaching walk if v only appears at the end of W . Define Svi = {u : there exists a reaching PC walk from u to v with ending color i}. Theorem 7. Let G be an edge-colored graph containing no PC cycle. Then one of the following holds. (1) G is semi-degenerate with δ c (G) ≤ 1; (2) G is non-degenerate and there exists a vertex z such that for each component F of G − z, F is joint to z by at most one color. Moreover, if col(z , F ) = {i}, then for each edge e in F , there exists a (z , i) PC walk contained in F + z and passing through e. Proof of Theorem 7. Without loss of generality, assume that G is connected with |G| ≥ 2 and contains no loops or multiple edges (since a loop does not effect the existence of PC cycles and two differently colored edges joining a same pair of vertices form a PC cycle of length 2, which contradicts that G is PC acyclic). If G is semi-degenerate, then let S be a degenerate set of G and let f be a function compatible to (G, S). Construct a digraph H with V (H) = S and A(H) = {uv : col(uv ) = f (u) and col(uv ) ̸ = f (v )}. 4 In 1983, Grossman and Häggkvist [7] firstly obtained the structure for 2-colored graphs. In 1997, Yeo [13] extended the result to edge-colored graphs of arbitrary number of colors. Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.
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Since G contains no PC cycle, H has no directed cycle. Thus there exists a vertex v ∈ S such that d− H (v ) = 0. By the construction of H and the definition of degenerate set, we can see that col(v, G − v ) = {f (v )}. Thus δ c (G) ≤ 1. Now assume that G is non-degenerate. Then by Theorem 4 (a), dw (u) ≥ 2 for each u ∈ V (G). By Theorem 6, let z be a vertex in G such that no component of G − z is joined to z with edges of more than one color. Let F be a component of G − z with col(z , F ) = {i}. First we prove that each vertex in F has a reaching PC walk to z. It suffices to show that if u ∈ V (F ) ∩ Szi , then each w ∈ NF (u) is contained in Szi . Let ueWhz be a reaching PC walk from u to z with ending color i. Let e′ be an edge joining w and u. If w is contained in W or col(e′ ) ̸ = col(e), then there is nothing to prove. So we may assume that w is not contained in W and col(e′ ) = col(e). Since dw (u) ≥ 2, G contains a (u, α ) PC walk ue1 W ′ e2 u such that α ̸ = col(e). If z is not contained in W ′ , then w e′ ue1 W ′ e2 ueWhz is a reaching PC walk from w to z with ending color i. Otherwise, we can find a reaching PC walk in W ′ from u to z such that the starting color is col(e1 ) and ending color is i (since col(z , F ) = {i}). Together with the edge e′ , we can see that w ∈ Szi . Now let e be an arbitrary edge in F joining u and w . We assert that there exist two reaching PC walks: W from u to z and W ′ from w to z such that neither of the starting colors is col(e). Because, from above, we can always obtain two reaching PC walks to z from u and w . Then consider a (u, α ) PC walk with α ̸ = col(e) and a (w, β ) PC walk with β ̸ = col(e), and revising the two reaching PC walks to z from u and w when it is necessary. The proof is complete. □ Remark 2. By Observation 1, more efforts should be paid to non-degenerate edge-colored graphs when studying PC cycle related problems. For instance, the minimum color degree condition given by Fujita and Magnant [5] for the existence of PC Hamilton cycles is true when the edge-colored complete graph is semi-degenerate and might be improved in the non-degenerate case. CRediT authorship contribution statement Ruonan Li: Conceptualization, Methodology, Writing - original draft. Binlong Li: Methodology. Shenggui Zhang: Writing - review & editing. Acknowledgments The authors are grateful to the referees for their detailed comments and valuable suggestions that greatly improved this article. The research is supported by NSFC (Nos. 11601429, 11671320 and 11901459), and the Fundamental Research Funds for the Central Universities of China (Nos. 31020180QD124 and 3102019GHJD003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.A. Bondy, U.S.R. Murty, Graph Theory, in: Springer Graduate Texts in Mathematics, vol. 244, 2008. R. Čada, K. Ozeki, K. Yoshimoto, A complete bipartite graph without properly colored cycles of length four, J. Graph Theory 93 (2020) 168–180. S. Fujita, R. Li, G. Wang, Decomposing edge-colored graphs under color degree constraints, Combin. Probab. Comput. 28 (2019) 755–767. S. Fujita, R. Li, S. Zhang, Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs, J. Graph Theory 87 (2018) 362–373. S. Fujita, C. Magnant, Properly colored paths and cycles, Discrete Appl. Math. 159 (2011) 1391–1397. T. Gallai, Transitiv orientierbare graphen, Acta Math. Acad. Sci. Hungar. 18 (1967) 25–66. J.W. Grossman, R. Häggkvist, Alternating cycles in edge-partitioned graphs, J. Combin. Theory Ser. B 34 (1983) 77–81. G. Gutin, M. Jones, B. Sheng, M. Wahlström, A. Yeo, Acyclicity in edge-colored graphs, Discrete Math. 340 (2017) 1–8. G. Gutin, M. Jones, B. Sheng, M. Wahlström, A. Yeo, Chinese postman problem on edge-colored multigraphs, Discrete Appl. Math. 217 (2017) 196–202. R. Li, H. Broersma, S. Zhang, Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs, J. Graph Theory, https://doi.org/10. 1002/jgt.22536. R. Li, H. Broersma, S. Zhang, Properly edge-colored theta graphs in edge-colored complete graphs, Graphs Combin. 35 (2019) 261–286. G. Wang, H. Li, Color degree and alternating cycles in edge-colored graphs, Discrete Math. 309 (2009) 4349–4354. A. Yeo, A note on alternating cycles in edge-colored graphs, J. Combin. Theory Ser. B 69 (1997) 222–225.
Please cite this article as: R. Li, B. Li and S. Zhang, A classification of edge-colored graphs based on properly colored walks, Discrete Applied Mathematics (2020), https://doi.org/10.1016/j.dam.2020.02.008.