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On the additive chromatic number of several families of graphs
Information Processing Letters 158 (2020) 105937
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
On the additive chromatic number of several families of graphs Daniel Severín ∗ Universidad Nacional de Rosario, Argentina CONICET, Argentina
a r t i c l e
i n f o
Article history: Received 15 November 2016 Received in revised form 13 February 2020 Accepted 16 February 2020 Available online 26 February 2020 Communicated by Marek Chrobak Keywords: Additive chromatic number Additive coloring conjecture Lucky labeling Graph algorithms
a b s t r a c t The Additive Coloring Problem is a variation of the Coloring Problem where labels of {1, . . . , k} are assigned to the vertices of a graph G so that the sum of labels over the neighborhood of each vertex is a proper coloring of G. The least value k for which G admits such labeling is called additive chromatic number of G. This problem was first presented by ˙ ´ Czerwinski, Grytczuk and Zelazny who also proposed a conjecture that for every graph G, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete graphs, trees, non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. In this work, we show that the conjecture holds for split graphs. We also present exact formulas for computing the additive chromatic number for some subfamilies of split graphs (complete split, headless spiders and complete sun), regular bipartite, complete multipartite, fan, windmill, circuit, wheel, cycle sun and wheel sun. © 2020 Elsevier B.V. All rights reserved.
1. Introduction The Additive Coloring Problem, also known as Lucky La´ Grytczuk beling Problem, was first presented by Czerwinski, ˙ and Zelazny in 2009 [5]. They also proposed the following conjecture: Additive Coloring Conjecture. [5] For every graph G, η ( G ) ≤ χ ( G ). Here, χ (G ) is the chromatic number of G and η(G ) is the additive chromatic number of G, defined below. For a given integer k, denote the set {1, 2, . . . , k} with [k]. Let G = ( V , E ) be a finite, undirected and simple
*
Correspondence to: Depto. de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Rosario, Pellegrini 250, Rosario, Argentina. E-mail address: [email protected]. https://doi.org/10.1016/j.ipl.2020.105937 0020-0190/© 2020 Elsevier B.V. All rights reserved.
graph, and f : V → [k] be a labeling of the vertices of G. Denote by f ( S ) the sum of labels over a set S ⊂ V , i.e. f ( S ) = u ∈ S f (u ). A labeling is an additive k-coloring if f ( N (u )) = f ( N ( v )) for all edges (u , v ) ∈ E, where N ( v ) is the open neighborhood of v. The additive chromatic number of G is defined as the least number k for which G has an additive k-coloring f , and is denoted by η(G ). The Additive Coloring Problem (ACP) consists of finding such number and is N P -hard [1]. In an attempt to prove the conjecture, several authors gave upper bounds of the additive chromatic number. For general graphs G with maximum degree , Akbari et al. proved that η(G ) ≤ 2 − + 1 [2]. For specific families of graphs, we have: if G is a tree, η(G ) ≤ 2 [5]; if G is planar bipartite, η(G ) ≤ 3 [5]; if G is planar, η(G ) ≤ 468 [3]; if G is 3-colorable and planar, η(G ) ≤ 36 [3]; if G is planar with girth at least 13, η(G ) ≤ 4 [3]; if G is planar with girth at least 26, η(G ) ≤ 3 [4].
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D. Severín / Information Processing Letters 158 (2020) 105937
Up to date, the conjecture has been proved for trees, complete graphs (since η( K n ) = n [5]), non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. Our contribution in this work is to give the exact value of the additive chromatic number of several families of graphs and expand the number of cases in which the conjecture is satisfied. In addition, we propose a tool which is used for checking the conjecture over all connected graphs up to 10 vertices. Now, we give some notation used throughout the article. For each v ∈ V , let N G ( v ) be the set of neighbors of v and d G ( v ) its degree, i.e. d G ( v ) = | N G ( v )|. Also, N G [ v ] = N G ( v ) ∪ { v }. If u , v are vertices of G, we say that u and v are true twins if N G [u ] = N G [ v ]. Let D = ( V , A ) be a finite directed graph. For each v ∈ V , define N D ( v ) = {(u , v ) ∈ A : u ∈ V } ∪ {( v , w ) ∈ A : w ∈ V }. When the graph or digraph is inferred from the context, we omit the subindex, i.e. d( v ), N ( v ), N [ v ]. Let D = ( V , A ) be a directed acyclic graph and G ( D ) be the undirected underlying graph of D. We say that D represents an acyclic orientation of G if G ( D ) is isomorphic to G. Let f : V → [k] be a labeling of vertices of D. If f ( N (u )) < f ( N ( v )) for every (u , v ) ∈ A, then f is called topological additive k-numbering of D. The topological additive number of D, denoted by ηt ( D ), is defined as the least number k for which D has a topological additive knumbering, or +∞ in case that such k does not exist. Clearly, η(G ) = min{ηt ( D ) : D represents an acyclic orientation of G }. Due to lack of space, we omit the proofs of Propositions. They can be found in an appendix [7] (also at https://arxiv.org/abs/1602.07675). 2. Regular bipartite and complete multipartite graphs As far as we know, the conjecture has not been proved for general bipartite graphs yet. We show that the conjecture holds for a subclass of bipartite graphs including regular ones. Observation 1. Let G = ( V , E ) be a graph, then and only if d(u ) = d( v ) for all (u , v ) ∈ E.
η(G ) = 1 if
Lemma 1. Let G = (U ∪ V , E ) be a bipartite graph (U and V are its stable sets) such that, for all v ∈ V and u ∈ N ( v ), d(u ) < 2d( v ). If d(u ) = d( v ) for all (u , v ) ∈ E then η(G ) = 1, otherwise η(G ) = 2. Proof. In virtue of Observation 1, we only have to prove η(G ) ≤ 2. Consider the assignment f : V → {1, 2} such that f (u ) = 2 for all u ∈ U and f ( v ) = 1 for all v ∈ V . Then, f ( N (u )) = d(u ) < 2d( v ) = f ( N ( v )) for all (u , v ) ∈ E. Corollary 1. If G is a regular bipartite graph, then η(G ) = 2. Now, we consider complete multipartite graphs. We say that a digraph D is complete r-partite when G ( D ) is complete r-partite. We say that D is monotone when V ( D ) can be partitioned into subsets V 1 , V 2 , . . . , V r such that every
arc in V i × V j satisfies i < j. In order to prove the theorem, we first cite a result given in [6]: Lemma 2. [6] Let D be a complete r-partite digraph. Then, ηt ( D ) < +∞ if and only D is monotone. In that case,
ηt ( D ) = max
si
|V i |
: i ∈ [r ] ,
where V 1 , . . . , V r is the partition of V ( D ), sr = | V r | and si = max{1 + si +1 , | V i |} for all i ∈ [r − 1]. Theorem 1. Let G = ( V 1 ∪· · ·∪ V r , E ) be the complete r-partite graph (V 1 , . . ., V r are its stable sets) and | V i | ≥ | V i +1 | for all i ∈ [r − 1]. Then, η(G ) = max{ | Vsii | : i ∈ [r ]} where sr = | V r | and si = max{1 + si +1 , | V i |} for all i ∈ [r − 1]. Moreover, η(G ) ≤ r. Proof. Let D be the monotone digraph such that G ( D ) = G and the partition of V ( D ) is V 1 , V 2 , . . . , V r . We must prove that D represents the acyclic orientation of G that provides the lowest value of ηt ( D ). Let D be another digraph representing an acyclic orientation of G with ηt ( D ) < ∞. Therefore, D is a monotone complete r-partite digraph where G ( D ) is isomorphic to G and the partition of V ( D ) is V i = V p(i ) for all i ∈ [r ] where p : [r ] → [r ] is some permutation function. Define si and s i for D and D respectively as in Lemma 2. It is easy to verify that sequences {si }i ∈[r ] and {s i }i ∈[r ] are decreasing. We now prove that s i ≥ si for all i, by induction on i = r , r − 1, . . . , 1. For the case i = r the statement is straightforward. For i < r, suppose that s i < si . By inductive hypothesis, 1 + si +1 ≤ s i < | V i |. Since s i ≥ | V j | for any j ≥ i, then | V i | > | V j |. Therefore p is not the identity function, and there exists some k < i such that p−1 (k) ≥ i, implying that | V i | > | V p −1 (k) | = | V k |. This leads to a contradiction as the sets from { V i }i ∈[r ] are arranged by size in decreasing order. Thus, s i ≥ si for all i ∈ [r ]. Let i be an integer such that si /| V i | is maximum and I = {t ∈ [r ] : | V t | = | V i |}. Note that i is the minimum index of I . Let J = {t ∈ [r ] : | V t | = | V i |} and j be the minimum index of J . Due to the ordering in the cardinality of sets of V ( D ), i ≥ j. Hence, s j ≥ s i ≥ si . Since j ∈ J , | V j | = | V i | and
we obtain s j /| V j | ≥ si /| V i |. Therefore, ηt ( D ) ≥ s j /| V j | ≥ si /| V i | = ηt ( D ). Now, we show that η(G ) ≤ r. We first prove by induction on i that si ≤ | V i |(r − i + 1) for i = r , r − 1, . . . , 1. In first place, if i = r, clearly sr = | V r | = | V r |(r − r + 1). If i < r, just two cases are possible. If si = | V i |, clearly si ≤ | V i |(r − i + 1). Otherwise, si = 1 + si +1 . By the inductive hypothesis si +1 ≤ | V i +1 |(r − i ) and the fact that | V i | ≥ | V i +1 |, we obtain:
si = 1 + si +1 ≤ 1 + | V i +1 |(r − i ) ≤ | V i +1 |(r − i + 1)
≤ | V i |(r − i + 1).
s
Hence, | Vi | ≤ r − i + 1 ≤ r for all i and therefore i η(G ) ≤ r. Since χ (G ) ≥ r for any complete r-partite graph G, we conclude that the conjecture holds for these graphs.
D. Severín / Information Processing Letters 158 (2020) 105937
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Let G 1 , G 2 be disjoint graphs. The join of G 1 with G 2 , denoted G 1 ∨ G 2 , is defined as the resulting graph G satisfying V (G ) = V (G 1 ) ∪ V (G 2 ) and E (G ) = E (G 1 ) ∪ E (G 2 ) ∪ {(u , v ) : u ∈ V (G 1 ), v ∈ V (G 2 )}. Given a graph G, the following theorem allows to solve the ACP of a join of G with a complete graph by just solving the ACP of G.
Let n, m be integers such that n ≥ 3, m ≥ 2. The windmill graph W nm is defined as m copies of K n which share a single vertex, i.e. W nm = mK n−1 ∨ K 1 . Then, η( W nm ) = n − 1. Let n be an integer such that n ≥ 4. A wheel is defined as W n = C n ∨ K 1 , where C n is a circuit of n vertices. First, we have to know η(C n ). If n is even, C n is regular bipartite. By Corollary 1, η( W n ) = η(C n ) = 2.
Observation 2. Let G = ( V , E ) be a graph and T ⊂ V such that any u , v ∈ T are true twins of G. Then, η(G ) ≥ | T |.
Proposition 1. Let n be an odd integer such that n ≥ 5. Then, η(C n ) = 3.
3. Join with complete graphs
Theorem 2. Let G be a graph of n vertices and be the largest degree in G. Then, η(G ∨ K q ) = max{η(G ), q} for all q ≤ n − − 1. Proof. Let V and E be the set of vertices and edges of G respectively, U = {u 1 , u 2 , . . . , u q } be the set of vertices of K q , G = G ∨ K q and f be an optimal additive coloring of G. Consider a labeling f of G satisfying f ( v ) = f ( v ) for all v ∈ V , and f (u i ) = i for all i ∈ [q]. Now, for any ( v , v ) ∈ E, f ( N G ( v )) = f ( N G ( v )) + f (U ) = f ( N G ( v )) + f (U ) = f ( N G ( v )). For any i , j ∈ [q] such that i < j, f ( N G (u i )) = f (U ∪ V ) − i > f (U ∪ V ) − j = f ( N G (u j )). Finally, note that f ( V \ N G ( v )) ≥ n − d G ( v ) ≥ n − for all v ∈ V . Then, for any u ∈ U and v ∈ V , f ( N G (u )) = f ( U ∪ V ) − f (u ) ≥ f ( U ∪ V ) − q > f ( U ∪ V ) − n + ≥ f (U ∪ V ) − f ( V \ N G ( v )) = f ( N G ( v )). Therefore, f is an additive coloring of G . In order to prove optimality, note first that any two vertices in U are true twins of G . By Observation 2, η(G ) ≥ q. In addition, suppose that η(G ) < η(G ). Hence, there exists an additive k-coloring f of G with k = η(G ) − 1. Let f be the labeling of G satisfying f ( v ) = f ( v ) for all v ∈ V . We have f ( N G ( v )) = f ( N G ( v )) − f (U ) = f ( N G ( v )) − f (U ) = f ( N G ( v )) for any ( v , v ) ∈ E. Therefore, f is an additive k-coloring of G which leads to a contradiction. When Theorem 2 is applied one must keep in mind that the size of a complete graph that can be joined to a graph is limited by n − − 1. In fact, if one chooses q = n − , η(G ∨ K q ) = max{η(G ), q} does no longer hold. For instance, consider the graph G ∗ of Fig. 1 and q = 2. It can be proven that η(G ∗ ) = 2 and η(G ∗ ∨ K 2 ) = 3. On the other hand, there are graphs G such that η(G ∨ K q ) = max{η(G ), q} for any q. An example is the family of stable graphs. In that case, G ∨ K q is called complete split. In the next section, we prove that the additive chromatic number of complete splits is q. The theorem also shows that if the conjecture holds for a graph G then it still holds for G ∨ K q (with q ≤ n − − 1) since χ (G ∨ K q ) = χ (G ) + q. A vertex v is universal in a graph G when N ( v ) = V (G )\{ v }. Note that, if G is a graph without universal vertices, the theorem asserts that η(G ∨ K 1 ) = η(G ). This is the case of fan, windmill and wheel graphs. Let n be an integer such that n ≥ 3. A n-fan is defined as F n = P n+1 ∨ K 1 where P n+1 is a path of length n. Since η( P n+1 ) = 2, η( F n ) = 2.
Therefore, if n is odd then
η( W n ) = 3.
4. Split graphs A graph G = ( V , E ) is a split graph if V can be partitioned in subsets Q , S such that Q is a clique of G and S is a stable set of G. We denote vertices of Q with u 1 , . . . , u q and vertices of S with v 1 , . . . , v s . W.l.o.g. we assume that Q is maximal (unless stated otherwise). The following result states an upper bound of the additive chromatic number of split graphs. Theorem 3. Let G = ( Q ∪ S , E ) be a split graph where Q is maximal and T ⊂ Q be a non-empty set such that the degrees of each vertex of T differ each other. Then, η(G ) ≤ | Q | − | T | + 1. Proof. W.l.o.g. let T = {u q−t +1 , u q−t +2 , . . . , u q−1 , u q } where t = | T |. We exhibit an additive (q − t + 1)-coloring of G. Consider the assignment f : V → [q − t + 1] such that f (u i ) = i for all i ∈ [q − t ], f ( w ) = q − t + 1 for all w ∈ T ∪ S. We first check for edges between the clique and the stable set. Let (u i , v ) ∈ E. Since Q is maximal, for each v ∈ S, there exists u ( v ) ∈ Q such that v is not adjacent to u ( v ). Then, f ( N ( v )) ≤ f ( Q ) − f (u ( v )) ≤ f ( Q ) − 1. On the other hand, let r i = | N (u i ) ∩ S | for all i ∈ [q]. Since v ∈ N (u i ), r i ≥ 1 and f ( N (u i )) = f ( Q ) − f (u i ) + (q − t + 1).r i ≥ f ( Q ). Therefore, f ( N (u i )) > f ( N ( v )). Now, we check for edges into the clique. First consider an edge (u j , uk ) such that u j , uk ∈ T . Then, r j = rk and f ( N (u j )) = f ( Q ) − (q − t + 1) + (q − t + 1).r j = f ( Q ) − (q − t + 1) + (q − t + 1).rk = f ( N (uk )). Finally consider an edge (u j , uk ) such that j ∈ [q − t ] and j < k. Let α = f (uk ) − f (u j ). Note that 1 ≤ α ≤ q − t. Then, f ( N (u j )) − f ( N (uk )) = α +(q − t + 1).(r j − rk ). Suppose that (q − t + 1).(r j − rk ) = α . Hence, 1 ≤ (q − t + 1).(r j − rk ) ≤ q − t. This contradicts r j − rk ∈ Z. Therefore, f ( N (u j )) = f ( N (uk )). Observe that η(G ) ≤ | Q | ≤ χ (G ), so the conjecture holds for split graphs. Now, we will see some subfamilies of split graphs where the exact value of η(G ) can be computed directly. A complete split is a graph G = ( Q ∪ S , E ) with | Q | ≥ 1, | S | ≥ 2, Q is a clique of G and there are edges (u , v ) for all u ∈ Q and v ∈ S . In these graphs, the bound given in Theorem 3 is tight.
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D. Severín / Information Processing Letters 158 (2020) 105937
Fig. 1. Some graphs: G ∗ , thin spider and wheel sun of order 5.
Proposition 2. Let G = ( Q ∪ S , E ) be a complete split. Then, η ( G ) = | Q |. For the next families, we use a result given in [6]: Lemma 3. [6] Let D = ( V , A ) be a directed acyclic graph such that its vertices are ordered so that (u , v ) ∈ A implies u < v. If Q is a clique of G ( D ) and q F , q L are the smallest and largest vertices of Q respectively, then
ηt ( D ) ≥
d(q F ) + 1 d(q L ) − | Q | + 2
.
5. Other suns In this section, we study cycle suns C S m , i.e. when the sun is obtained from a circuit (V (G ) = U and E (G ) = {(u i , u i −1 ) : i ∈ [m]}), and wheel suns W S m , i.e. when the sun is obtained from a wheel (V (G ) = U ∪ { w } and E (G ) = {(u i , u i −1 ), (u i , w ) : i ∈ [m]}). Fig. 1 displays a wheel graph with m = 5. Proposition 5. Let m ≥ 4. Then, η(C S m ) = η( W S m ) = 2. Clearly, the conjecture is satisfied in these graphs. 6. A tool for solving the ACP
Corollary 2. Let G be a graph and Q be a clique of G. If d1 , d2 are the degrees of the vertices of Q with smallest and largest degree respectively, then
η(G ) ≥
d1 + 1 d2 − | Q | + 2
.
A thin headless spider of order q ≥ 2 is a split graph where | Q | = | S | = q and the set of edges between Q and S is {(u i , v i ) : i ∈ [q]}. Fig. 1 shows an example of a thin spider of order 5. A thick headless spider of order q ≥ 2 is a split graph where | Q | = | S | = q and the set of edges between Q and S is {(u i , v j ) : i , j ∈ [q], i = j }. Equivalently, a thick headless spider is the complement of a thin headless spider of the same order and vice-versa. Proposition 3. Let G be a thin/thick headless spider of order q. Then,
η(G ) =
q+1 2
.
Let G be a graph and U = {u 1 , . . . , um } ⊂ V (G ). A sun is a graph G obtained from G as follows: V (G ) = V (G ) ∪ V where V = { v 1 , . . . , v m } and
E (G ) = E (G ) ∪ {(u i , v i −1 ), (u i , v i ) : i ∈ [m]}.
m+2 3
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was partially supported by grants PID-ING 416 (UNR), PICT-2013-0586 (MINCyT) and PIP 11220120100277 (CONICET). I wish to thank Dr. Graciela Nasini for their helpful comments. Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.ipl.2020.105937. References
For the sake of simplicity, u 0 and v 0 are another names for um and v m . A complete sun of order m, denoted by K S m , is a split graph obtained from a complete graph G of size m. Proposition 4. Let m ≥ 3. Then, η( K S m ) =
As far as we know, there are no tools available for solving ACP. However, we can solve instances of this problem by modeling it as an integer linear programming formulation and using an available solver like CPLEX. The source code of such tool can be downloaded from [7]. Besides this tool has been very useful for checking our theoretical results, we have tested the conjecture over all connected graphs up to 10 vertices (about 12 million graphs).
.
[1] A. Ahadi, A. Dehghan, M. Kazemi, E. Mollaahmadi, Computation of lucky number of planar graphs is NP-hard, Inf. Process. Lett. 112 (2012) 109–112. [2] S. Akbari, M. Ghanbari, R. Manaviyat, S. Zare, On the lucky choice number of graphs, Graphs Comb. 29 (2013) 157–163.
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´ [3] T. Bartnicki, B. Bosek, S. Czerwinski, J. Grytczuk, G. Matecki, W. ˙ Additive colorings of planar graphs, Graphs Comb. 30 (2014) Zelazny, 1087–1098. [4] A. Brandt, S. Jahanbekam, J. White, Additive list coloring of planar graphs with given girth, Discuss. Math., Graph Theory (2020) 1–19, https://doi.org/10.7151/dmgt.2156, in press. ˙ ´ [5] S. Czerwinski, J. Grytczuk, W. Zelazny, Lucky labelings of graphs, Inf. Process. Lett. 109 (2009) 1078–1081.
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[6] J. Marenco, M. Mydlarz, D. Severín, Topological additive numbering of directed acyclic graphs, Inf. Process. Lett. 115 (2015) 199–202. [7] D. Severín, Appendix and source code of ACP solver for the paper on the additive chromatic number of several families of graphs, Mendeley Data, v1, 2020, https://doi.org/10.17632/9zwm2nxvbs.1.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
On the additive chromatic number of several families of graphs Daniel Severín ∗ Universidad Nacional de Rosario, Argentina CONICET, Argentina
a r t i c l e
i n f o
Article history: Received 15 November 2016 Received in revised form 13 February 2020 Accepted 16 February 2020 Available online 26 February 2020 Communicated by Marek Chrobak Keywords: Additive chromatic number Additive coloring conjecture Lucky labeling Graph algorithms
a b s t r a c t The Additive Coloring Problem is a variation of the Coloring Problem where labels of {1, . . . , k} are assigned to the vertices of a graph G so that the sum of labels over the neighborhood of each vertex is a proper coloring of G. The least value k for which G admits such labeling is called additive chromatic number of G. This problem was first presented by ˙ ´ Czerwinski, Grytczuk and Zelazny who also proposed a conjecture that for every graph G, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete graphs, trees, non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. In this work, we show that the conjecture holds for split graphs. We also present exact formulas for computing the additive chromatic number for some subfamilies of split graphs (complete split, headless spiders and complete sun), regular bipartite, complete multipartite, fan, windmill, circuit, wheel, cycle sun and wheel sun. © 2020 Elsevier B.V. All rights reserved.
1. Introduction The Additive Coloring Problem, also known as Lucky La´ Grytczuk beling Problem, was first presented by Czerwinski, ˙ and Zelazny in 2009 [5]. They also proposed the following conjecture: Additive Coloring Conjecture. [5] For every graph G, η ( G ) ≤ χ ( G ). Here, χ (G ) is the chromatic number of G and η(G ) is the additive chromatic number of G, defined below. For a given integer k, denote the set {1, 2, . . . , k} with [k]. Let G = ( V , E ) be a finite, undirected and simple
*
Correspondence to: Depto. de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Rosario, Pellegrini 250, Rosario, Argentina. E-mail address: [email protected]. https://doi.org/10.1016/j.ipl.2020.105937 0020-0190/© 2020 Elsevier B.V. All rights reserved.
graph, and f : V → [k] be a labeling of the vertices of G. Denote by f ( S ) the sum of labels over a set S ⊂ V , i.e. f ( S ) = u ∈ S f (u ). A labeling is an additive k-coloring if f ( N (u )) = f ( N ( v )) for all edges (u , v ) ∈ E, where N ( v ) is the open neighborhood of v. The additive chromatic number of G is defined as the least number k for which G has an additive k-coloring f , and is denoted by η(G ). The Additive Coloring Problem (ACP) consists of finding such number and is N P -hard [1]. In an attempt to prove the conjecture, several authors gave upper bounds of the additive chromatic number. For general graphs G with maximum degree , Akbari et al. proved that η(G ) ≤ 2 − + 1 [2]. For specific families of graphs, we have: if G is a tree, η(G ) ≤ 2 [5]; if G is planar bipartite, η(G ) ≤ 3 [5]; if G is planar, η(G ) ≤ 468 [3]; if G is 3-colorable and planar, η(G ) ≤ 36 [3]; if G is planar with girth at least 13, η(G ) ≤ 4 [3]; if G is planar with girth at least 26, η(G ) ≤ 3 [4].
2
D. Severín / Information Processing Letters 158 (2020) 105937
Up to date, the conjecture has been proved for trees, complete graphs (since η( K n ) = n [5]), non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. Our contribution in this work is to give the exact value of the additive chromatic number of several families of graphs and expand the number of cases in which the conjecture is satisfied. In addition, we propose a tool which is used for checking the conjecture over all connected graphs up to 10 vertices. Now, we give some notation used throughout the article. For each v ∈ V , let N G ( v ) be the set of neighbors of v and d G ( v ) its degree, i.e. d G ( v ) = | N G ( v )|. Also, N G [ v ] = N G ( v ) ∪ { v }. If u , v are vertices of G, we say that u and v are true twins if N G [u ] = N G [ v ]. Let D = ( V , A ) be a finite directed graph. For each v ∈ V , define N D ( v ) = {(u , v ) ∈ A : u ∈ V } ∪ {( v , w ) ∈ A : w ∈ V }. When the graph or digraph is inferred from the context, we omit the subindex, i.e. d( v ), N ( v ), N [ v ]. Let D = ( V , A ) be a directed acyclic graph and G ( D ) be the undirected underlying graph of D. We say that D represents an acyclic orientation of G if G ( D ) is isomorphic to G. Let f : V → [k] be a labeling of vertices of D. If f ( N (u )) < f ( N ( v )) for every (u , v ) ∈ A, then f is called topological additive k-numbering of D. The topological additive number of D, denoted by ηt ( D ), is defined as the least number k for which D has a topological additive knumbering, or +∞ in case that such k does not exist. Clearly, η(G ) = min{ηt ( D ) : D represents an acyclic orientation of G }. Due to lack of space, we omit the proofs of Propositions. They can be found in an appendix [7] (also at https://arxiv.org/abs/1602.07675). 2. Regular bipartite and complete multipartite graphs As far as we know, the conjecture has not been proved for general bipartite graphs yet. We show that the conjecture holds for a subclass of bipartite graphs including regular ones. Observation 1. Let G = ( V , E ) be a graph, then and only if d(u ) = d( v ) for all (u , v ) ∈ E.
η(G ) = 1 if
Lemma 1. Let G = (U ∪ V , E ) be a bipartite graph (U and V are its stable sets) such that, for all v ∈ V and u ∈ N ( v ), d(u ) < 2d( v ). If d(u ) = d( v ) for all (u , v ) ∈ E then η(G ) = 1, otherwise η(G ) = 2. Proof. In virtue of Observation 1, we only have to prove η(G ) ≤ 2. Consider the assignment f : V → {1, 2} such that f (u ) = 2 for all u ∈ U and f ( v ) = 1 for all v ∈ V . Then, f ( N (u )) = d(u ) < 2d( v ) = f ( N ( v )) for all (u , v ) ∈ E. Corollary 1. If G is a regular bipartite graph, then η(G ) = 2. Now, we consider complete multipartite graphs. We say that a digraph D is complete r-partite when G ( D ) is complete r-partite. We say that D is monotone when V ( D ) can be partitioned into subsets V 1 , V 2 , . . . , V r such that every
arc in V i × V j satisfies i < j. In order to prove the theorem, we first cite a result given in [6]: Lemma 2. [6] Let D be a complete r-partite digraph. Then, ηt ( D ) < +∞ if and only D is monotone. In that case,
ηt ( D ) = max
si
|V i |
: i ∈ [r ] ,
where V 1 , . . . , V r is the partition of V ( D ), sr = | V r | and si = max{1 + si +1 , | V i |} for all i ∈ [r − 1]. Theorem 1. Let G = ( V 1 ∪· · ·∪ V r , E ) be the complete r-partite graph (V 1 , . . ., V r are its stable sets) and | V i | ≥ | V i +1 | for all i ∈ [r − 1]. Then, η(G ) = max{ | Vsii | : i ∈ [r ]} where sr = | V r | and si = max{1 + si +1 , | V i |} for all i ∈ [r − 1]. Moreover, η(G ) ≤ r. Proof. Let D be the monotone digraph such that G ( D ) = G and the partition of V ( D ) is V 1 , V 2 , . . . , V r . We must prove that D represents the acyclic orientation of G that provides the lowest value of ηt ( D ). Let D be another digraph representing an acyclic orientation of G with ηt ( D ) < ∞. Therefore, D is a monotone complete r-partite digraph where G ( D ) is isomorphic to G and the partition of V ( D ) is V i = V p(i ) for all i ∈ [r ] where p : [r ] → [r ] is some permutation function. Define si and s i for D and D respectively as in Lemma 2. It is easy to verify that sequences {si }i ∈[r ] and {s i }i ∈[r ] are decreasing. We now prove that s i ≥ si for all i, by induction on i = r , r − 1, . . . , 1. For the case i = r the statement is straightforward. For i < r, suppose that s i < si . By inductive hypothesis, 1 + si +1 ≤ s i < | V i |. Since s i ≥ | V j | for any j ≥ i, then | V i | > | V j |. Therefore p is not the identity function, and there exists some k < i such that p−1 (k) ≥ i, implying that | V i | > | V p −1 (k) | = | V k |. This leads to a contradiction as the sets from { V i }i ∈[r ] are arranged by size in decreasing order. Thus, s i ≥ si for all i ∈ [r ]. Let i be an integer such that si /| V i | is maximum and I = {t ∈ [r ] : | V t | = | V i |}. Note that i is the minimum index of I . Let J = {t ∈ [r ] : | V t | = | V i |} and j be the minimum index of J . Due to the ordering in the cardinality of sets of V ( D ), i ≥ j. Hence, s j ≥ s i ≥ si . Since j ∈ J , | V j | = | V i | and
we obtain s j /| V j | ≥ si /| V i |. Therefore, ηt ( D ) ≥ s j /| V j | ≥ si /| V i | = ηt ( D ). Now, we show that η(G ) ≤ r. We first prove by induction on i that si ≤ | V i |(r − i + 1) for i = r , r − 1, . . . , 1. In first place, if i = r, clearly sr = | V r | = | V r |(r − r + 1). If i < r, just two cases are possible. If si = | V i |, clearly si ≤ | V i |(r − i + 1). Otherwise, si = 1 + si +1 . By the inductive hypothesis si +1 ≤ | V i +1 |(r − i ) and the fact that | V i | ≥ | V i +1 |, we obtain:
si = 1 + si +1 ≤ 1 + | V i +1 |(r − i ) ≤ | V i +1 |(r − i + 1)
≤ | V i |(r − i + 1).
s
Hence, | Vi | ≤ r − i + 1 ≤ r for all i and therefore i η(G ) ≤ r. Since χ (G ) ≥ r for any complete r-partite graph G, we conclude that the conjecture holds for these graphs.
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Let G 1 , G 2 be disjoint graphs. The join of G 1 with G 2 , denoted G 1 ∨ G 2 , is defined as the resulting graph G satisfying V (G ) = V (G 1 ) ∪ V (G 2 ) and E (G ) = E (G 1 ) ∪ E (G 2 ) ∪ {(u , v ) : u ∈ V (G 1 ), v ∈ V (G 2 )}. Given a graph G, the following theorem allows to solve the ACP of a join of G with a complete graph by just solving the ACP of G.
Let n, m be integers such that n ≥ 3, m ≥ 2. The windmill graph W nm is defined as m copies of K n which share a single vertex, i.e. W nm = mK n−1 ∨ K 1 . Then, η( W nm ) = n − 1. Let n be an integer such that n ≥ 4. A wheel is defined as W n = C n ∨ K 1 , where C n is a circuit of n vertices. First, we have to know η(C n ). If n is even, C n is regular bipartite. By Corollary 1, η( W n ) = η(C n ) = 2.
Observation 2. Let G = ( V , E ) be a graph and T ⊂ V such that any u , v ∈ T are true twins of G. Then, η(G ) ≥ | T |.
Proposition 1. Let n be an odd integer such that n ≥ 5. Then, η(C n ) = 3.
3. Join with complete graphs
Theorem 2. Let G be a graph of n vertices and be the largest degree in G. Then, η(G ∨ K q ) = max{η(G ), q} for all q ≤ n − − 1. Proof. Let V and E be the set of vertices and edges of G respectively, U = {u 1 , u 2 , . . . , u q } be the set of vertices of K q , G = G ∨ K q and f be an optimal additive coloring of G. Consider a labeling f of G satisfying f ( v ) = f ( v ) for all v ∈ V , and f (u i ) = i for all i ∈ [q]. Now, for any ( v , v ) ∈ E, f ( N G ( v )) = f ( N G ( v )) + f (U ) = f ( N G ( v )) + f (U ) = f ( N G ( v )). For any i , j ∈ [q] such that i < j, f ( N G (u i )) = f (U ∪ V ) − i > f (U ∪ V ) − j = f ( N G (u j )). Finally, note that f ( V \ N G ( v )) ≥ n − d G ( v ) ≥ n − for all v ∈ V . Then, for any u ∈ U and v ∈ V , f ( N G (u )) = f ( U ∪ V ) − f (u ) ≥ f ( U ∪ V ) − q > f ( U ∪ V ) − n + ≥ f (U ∪ V ) − f ( V \ N G ( v )) = f ( N G ( v )). Therefore, f is an additive coloring of G . In order to prove optimality, note first that any two vertices in U are true twins of G . By Observation 2, η(G ) ≥ q. In addition, suppose that η(G ) < η(G ). Hence, there exists an additive k-coloring f of G with k = η(G ) − 1. Let f be the labeling of G satisfying f ( v ) = f ( v ) for all v ∈ V . We have f ( N G ( v )) = f ( N G ( v )) − f (U ) = f ( N G ( v )) − f (U ) = f ( N G ( v )) for any ( v , v ) ∈ E. Therefore, f is an additive k-coloring of G which leads to a contradiction. When Theorem 2 is applied one must keep in mind that the size of a complete graph that can be joined to a graph is limited by n − − 1. In fact, if one chooses q = n − , η(G ∨ K q ) = max{η(G ), q} does no longer hold. For instance, consider the graph G ∗ of Fig. 1 and q = 2. It can be proven that η(G ∗ ) = 2 and η(G ∗ ∨ K 2 ) = 3. On the other hand, there are graphs G such that η(G ∨ K q ) = max{η(G ), q} for any q. An example is the family of stable graphs. In that case, G ∨ K q is called complete split. In the next section, we prove that the additive chromatic number of complete splits is q. The theorem also shows that if the conjecture holds for a graph G then it still holds for G ∨ K q (with q ≤ n − − 1) since χ (G ∨ K q ) = χ (G ) + q. A vertex v is universal in a graph G when N ( v ) = V (G )\{ v }. Note that, if G is a graph without universal vertices, the theorem asserts that η(G ∨ K 1 ) = η(G ). This is the case of fan, windmill and wheel graphs. Let n be an integer such that n ≥ 3. A n-fan is defined as F n = P n+1 ∨ K 1 where P n+1 is a path of length n. Since η( P n+1 ) = 2, η( F n ) = 2.
Therefore, if n is odd then
η( W n ) = 3.
4. Split graphs A graph G = ( V , E ) is a split graph if V can be partitioned in subsets Q , S such that Q is a clique of G and S is a stable set of G. We denote vertices of Q with u 1 , . . . , u q and vertices of S with v 1 , . . . , v s . W.l.o.g. we assume that Q is maximal (unless stated otherwise). The following result states an upper bound of the additive chromatic number of split graphs. Theorem 3. Let G = ( Q ∪ S , E ) be a split graph where Q is maximal and T ⊂ Q be a non-empty set such that the degrees of each vertex of T differ each other. Then, η(G ) ≤ | Q | − | T | + 1. Proof. W.l.o.g. let T = {u q−t +1 , u q−t +2 , . . . , u q−1 , u q } where t = | T |. We exhibit an additive (q − t + 1)-coloring of G. Consider the assignment f : V → [q − t + 1] such that f (u i ) = i for all i ∈ [q − t ], f ( w ) = q − t + 1 for all w ∈ T ∪ S. We first check for edges between the clique and the stable set. Let (u i , v ) ∈ E. Since Q is maximal, for each v ∈ S, there exists u ( v ) ∈ Q such that v is not adjacent to u ( v ). Then, f ( N ( v )) ≤ f ( Q ) − f (u ( v )) ≤ f ( Q ) − 1. On the other hand, let r i = | N (u i ) ∩ S | for all i ∈ [q]. Since v ∈ N (u i ), r i ≥ 1 and f ( N (u i )) = f ( Q ) − f (u i ) + (q − t + 1).r i ≥ f ( Q ). Therefore, f ( N (u i )) > f ( N ( v )). Now, we check for edges into the clique. First consider an edge (u j , uk ) such that u j , uk ∈ T . Then, r j = rk and f ( N (u j )) = f ( Q ) − (q − t + 1) + (q − t + 1).r j = f ( Q ) − (q − t + 1) + (q − t + 1).rk = f ( N (uk )). Finally consider an edge (u j , uk ) such that j ∈ [q − t ] and j < k. Let α = f (uk ) − f (u j ). Note that 1 ≤ α ≤ q − t. Then, f ( N (u j )) − f ( N (uk )) = α +(q − t + 1).(r j − rk ). Suppose that (q − t + 1).(r j − rk ) = α . Hence, 1 ≤ (q − t + 1).(r j − rk ) ≤ q − t. This contradicts r j − rk ∈ Z. Therefore, f ( N (u j )) = f ( N (uk )). Observe that η(G ) ≤ | Q | ≤ χ (G ), so the conjecture holds for split graphs. Now, we will see some subfamilies of split graphs where the exact value of η(G ) can be computed directly. A complete split is a graph G = ( Q ∪ S , E ) with | Q | ≥ 1, | S | ≥ 2, Q is a clique of G and there are edges (u , v ) for all u ∈ Q and v ∈ S . In these graphs, the bound given in Theorem 3 is tight.
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Fig. 1. Some graphs: G ∗ , thin spider and wheel sun of order 5.
Proposition 2. Let G = ( Q ∪ S , E ) be a complete split. Then, η ( G ) = | Q |. For the next families, we use a result given in [6]: Lemma 3. [6] Let D = ( V , A ) be a directed acyclic graph such that its vertices are ordered so that (u , v ) ∈ A implies u < v. If Q is a clique of G ( D ) and q F , q L are the smallest and largest vertices of Q respectively, then
ηt ( D ) ≥
d(q F ) + 1 d(q L ) − | Q | + 2
.
5. Other suns In this section, we study cycle suns C S m , i.e. when the sun is obtained from a circuit (V (G ) = U and E (G ) = {(u i , u i −1 ) : i ∈ [m]}), and wheel suns W S m , i.e. when the sun is obtained from a wheel (V (G ) = U ∪ { w } and E (G ) = {(u i , u i −1 ), (u i , w ) : i ∈ [m]}). Fig. 1 displays a wheel graph with m = 5. Proposition 5. Let m ≥ 4. Then, η(C S m ) = η( W S m ) = 2. Clearly, the conjecture is satisfied in these graphs. 6. A tool for solving the ACP
Corollary 2. Let G be a graph and Q be a clique of G. If d1 , d2 are the degrees of the vertices of Q with smallest and largest degree respectively, then
η(G ) ≥
d1 + 1 d2 − | Q | + 2
.
A thin headless spider of order q ≥ 2 is a split graph where | Q | = | S | = q and the set of edges between Q and S is {(u i , v i ) : i ∈ [q]}. Fig. 1 shows an example of a thin spider of order 5. A thick headless spider of order q ≥ 2 is a split graph where | Q | = | S | = q and the set of edges between Q and S is {(u i , v j ) : i , j ∈ [q], i = j }. Equivalently, a thick headless spider is the complement of a thin headless spider of the same order and vice-versa. Proposition 3. Let G be a thin/thick headless spider of order q. Then,
η(G ) =
q+1 2
.
Let G be a graph and U = {u 1 , . . . , um } ⊂ V (G ). A sun is a graph G obtained from G as follows: V (G ) = V (G ) ∪ V where V = { v 1 , . . . , v m } and
E (G ) = E (G ) ∪ {(u i , v i −1 ), (u i , v i ) : i ∈ [m]}.
m+2 3
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was partially supported by grants PID-ING 416 (UNR), PICT-2013-0586 (MINCyT) and PIP 11220120100277 (CONICET). I wish to thank Dr. Graciela Nasini for their helpful comments. Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.ipl.2020.105937. References
For the sake of simplicity, u 0 and v 0 are another names for um and v m . A complete sun of order m, denoted by K S m , is a split graph obtained from a complete graph G of size m. Proposition 4. Let m ≥ 3. Then, η( K S m ) =
As far as we know, there are no tools available for solving ACP. However, we can solve instances of this problem by modeling it as an integer linear programming formulation and using an available solver like CPLEX. The source code of such tool can be downloaded from [7]. Besides this tool has been very useful for checking our theoretical results, we have tested the conjecture over all connected graphs up to 10 vertices (about 12 million graphs).
.
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