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Nowhere-zero Z3 -flows through Z3 -connectivity
Discrete Mathematics 306 (2006) 26 – 30 www.elsevier.com/locate/disc
Nowhere-zero Z3-flows through Z3-connectivity Matt DeVosa , Rui Xub,∗ , Gexin Yuc a Department of Mathematics, Princeton University, Princeton, NJ 08544, USA b Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA c Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 2 September 2005; received in revised form 14 October 2005; accepted 21 October 2005 Available online 27 December 2005
Abstract Let be an abelian group. Jaeger et al. [Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182] introduced a class of graphs which they call -connected. The main interest in -connected graphs is that every -connected graph admits a nowhere-zero -flow. In this paper, we found some families of Z3 -connected graphs. Our results generalize an early theorem by Lai (Nowhere-zero 3-flows in locally connected graphs, J. Graph Theory 42 (2003) 211–219) for nowhere-zero Z3 -flows in locally connected graphs, and provide a simplified proof of a theorem of Xu and Zhang on nowhere-zero Z3 -flows in squares of graphs. © 2005 Elsevier B.V. All rights reserved. Keywords: Integer flow; Group connectivity; Squares of graphs
1. Introduction Throughout this article, graphs and directed graphs may have multiple edges and loops. Let G be a directed graph, let be an abelian group and let : E(G) → be a map. is nowhere-zero if 0 ∈ / (E(G)). The boundary We say that of is the map j : V (G) → where j(v) = e∈E + (v) (e) − e∈E − (v) (e) for each vertex v ∈ V (G) (note that v∈V (G) j(v) = 0), where E + (v) (or E − (v)) is the set of all arcs of G with their tails (or, heads, respectively) at the vertex v. If j is identically zero, then is called a flow or a -flow. We say that G is -connected if, for every p : V (G) → with v∈V (G) p(v) = 0, there exists a nowhere-zero map : E(G) → with boundary j = p. If we change the orientation of the edge e and switch (e) to −(e), then the boundary is preserved, and the new map is nowhere-zero if and only if the original was. Thus the existence of a nowhere-zero map with a specified boundary depends only on the underlying undirected graph and not on the orientation of the edges. Accordingly, we say that an undirected graph admits a nowhere-zero -flow (is -connected) if some (and thus every) orientation of it admits a nowhere-zero -flow (is -connected). In the remainder of the paper, we will restrict our attention to the group Z3 = Z/3Z. The main interest in nowhere-zero Z3 -flows stems from the following conjecture of Tutte.
∗ Corresponding author. Tel.: +1 678 839 4122; fax: +1 678 839 6490.
E-mail addresses: [email protected] (M. DeVos), [email protected] (R. Xu), [email protected] (G. Yu). 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.10.019
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Conjecture 1.1 (The 3-flow conjecture, Tutte; see unsolved problem 48 in [1]). Every 4-edge-connected graph admits a nowhere-zero Z3 -flow. In their article on group connectivity, Jaeger et al. make the following conjecture on Z3 -connected graphs. Conjecture 1.2 (Jaeger et al. [2]). Every 5-edge connected graph is Z3 -connected. Recently Kochol [3] has shown that Conjecture 1.1 can be reduced to 5-edge-connected graphs. It follows from this that Conjecture 1.2 implies Conjecture 1.1. Thus, it may be possible to achieve the 3-flow conjecture by instead proving a result on Z3 -connectivity. Although we appear to be quite far from a resolution to either of these conjectures, the idea of using Z3 -connectivity to establish results on Z3 -flows is quite useful. Indeed, this is the main theme of this paper. Before stating our main theorems, we shall introduce a definition and an easy observation which is also suggestive of this theme. If H is a connected subgraph of the graph G, then G contracts H, denoted by G/H , is the graph obtained from G by deleting all edges in H and then identifying V (H ) to a single new vertex. The following observation follows easily from the definitions. Observation 1.3. Let H be a Z3 -connected subgraph of the graph G. (i) If G/H admits a nowhere-zero Z3 -flow, then so does G. (ii) If G/H is Z3 -connected, then so is G. Let us call a Z3 -connected graph on > 1 vertex reducible. Now suppose that F is a class of graphs which is closed under contracting subgraphs (for instance the class of all 5-edge-connected graphs). In order to prove that every graph in F admits a nowhere-zero Z3 -flow, it suffices to consider only those graphs which do not contain a reducible subgraph. Similarly, to prove that every graph in F is Z3 -connected, it suffices to show that every graph in F contains a reducible subgraph. This is suggestive of a kind of reducibility/unavoidability approach to Conjecture 1.2. An obvious difficulty in this approach is that for any finite list of reducible graphs H1 , H2 , . . . , Hk , there exist 5-edge-connected graphs containing no subgraph isomorphic to Hi for 1 i k (this follows from the existence of 5-edge-connected graphs of high girth and the fact that every reducible graph contains a circuit). It is possible to prove that graphs with some added structure are Z3 -connected. A graph G is locally connected if G[N(v)] is connected for every v ∈ V (G). We define G to be triangle connected if for every e, f ∈ E(G), there exists a sequence of circuits C1 , C2 , . . . , Ck so that e ∈ E(C1 ), f ∈ E(Ck ), |E(Ci )| 3 for 1 i k, and E(Cj )∩E(Cj +1 ) = ∅ for 1 j k − 1. It is easy to see that every locally connected graph on at least three vertices is also triangle connected. The following theorem gives a sufficient condition for a graph to be Z3 -connected. Theorem 1.4. If G is a loopless triangle connected graph with minimum degree 4 then G is Z3 -connected. This theorem is a generalization of the following result of Lai. Theorem 1.5 (Lai [4]). If G is 2-edge-connected and G[N (v)] is 3-edge-connected for every v ∈ V (G) then G has a nowhere-zero Z3 -flow. To see that Theorem 1.4 implies Theorem 1.5, note that every graph G on > 2 vertices satisfying the input condition of Theorem 1.5 is 2-connected and triangle connected. If v ∈ V (G) and u ∈ N (v), then degG[N(u)] (v) 3, so degG (v) 4. Thus the result follows from Theorem 1.4. If G is a simple and loopless graph, then G2 is the graph with V (G2 ) = V (G) so that u, v ∈ V (G2 ) are adjacent (in 2 G ) if and only if they are distance 2 in G. Using our techniques, we establish necessary and sufficient conditions on G for G2 to be Z3 -connected. As a consequence of this, we obtain a simplified proof of a theorem of Xu and Zhang [5] which gives necessary and sufficient conditions on G for G2 to admit a nowhere-zero Z3 -flow. Before stating this results, we require the following definition.
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Definition 1.6. T1,3 = {T |T is a tree and dT (v) ∈ {1, 3} for every v ∈ V (T )}. ¯ 1,3 = {T |T ∈ T1,3 or T is a simple 4-vertex graph containing a 4-circuit or T is obtained from some T ∈ T1,3 T by adding some edges which join two distance 2 leaves of T }. T1,2,3 = {T |T is a tree and dT (v) ∈ {1, 2, 3} for every v ∈ V (T )}. ¯ 1,2,3 = {T |T ∈ T1,2,3 or T is a simple 4-vertex graph containing a 4-circuit or T is obtained from T some T ∈ T1,2,3 by adding some edges which join two distance 2 leaves of T }. ¯ 1,2,3 . Theorem 1.7. Let G be a simple connected graph. Then G2 is Z3 -connected if and only if G ∈ /T Theorem 1.7 is a generalization of the following result obtained recently by Xu and Zhang. Theorem 1.8 (Xu and Zhang [5]). Let G be a simple connected graph. Then G2 admits a nowhere-zero Z3 -flow if and ¯ 1,3 . only if G ∈ /T 2. Proofs We begin with the following observation which follows easily from the definitions. Observation 2.1. Let H be a Z3 -connected subgraph of the graph G. (i) If V (G) = V (H ), then G is Z3 -connected. (ii) If V (G)\V (H ) = {u} and there are at least two edges joining u to the vertex (or vertices) of V (H ), then G is Z3 -connected. For every integer k 2, let Ck denote a graph which is a circuit on k vertices. A special case of (ii) above is that the graph C2 is Z3 -connected. The proof of Theorem 1.4 requires the following easy proposition. Proposition 2.2. If G is triangle connected and H < G is a Z3 -connected graph on at least two vertices, then G is Z3 -connected. Proof. Choose a maximal Z3 -connected subgraph H of G with H < H . If V (H ) = V (G), then there must exist an edge e ∈ E(H ) and a triangle C < G so that e ∈ E(C) and V (C)V (H ). It now follows from (ii) of Observation 2.1 that H ∪ C is Z3 -connected, contradicting our choice. Proof of Theorem 1.4. If |V (G)| 3 then the theorem is obvious, so we shall assume that |V (G)| 4. Choose a maximal triangle connected subgraph H < G with V (H ) = V (G). Since G is triangle connected, it follows easily that V (G)\V (H ) contains a single vertex, say u. It now follows from our assumptions that we may choose an edge e ∈ E(H ) and a triangle C < G so that u ∈ V (C) and e ∈ E(C). Modify the graph G to form a new graph G by deleting E(C)\{e} and adding a new edge e parallel to e. Let H < G be obtained from H by adding the edge e . If G is Z3 -connected, then it follows easily that G is also Z3 -connected. Thus, the theorem follows from the fact that H is Z3 -connected (by Proposition 2.2 and the fact that C2 is Z3 -connected) and part (ii) of Observation 2.1. Corollary 2.3. If n 5, then Cn2 is Z3 -connected, where Cn is a circuit of length n. Next we will prove that another class of graphs is Z3 -connected. For every n 3, we define Wn to be a simple graph obtained from a circuit of length n by adding a new vertex adjacent to all existing vertices. Proposition 2.4. For every integer k 2, the graph W2k is Z3 -connected. Proof. Let G be a graph obtained from a circuit of length 2k by adding a new vertex x adjacent to all existing vertices. Let p : V (G) → Z3 be a map with v∈V (G) p(v) = 0. If p is identically zero, then it is easy to verify that G
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admits a nowhere-zero map with boundary p. Otherwise, we may choose a vertex y ∈ V (G)\{x} with p(y) = 0. Let N(y) = {x, z1 , z2 }, let G be the graph obtained from G\y by adding a new edge (in parallel) from z1 to x, and let p : V (G ) → Z3 be given by the rule p (v) = p(v) if v ∈ V (G )\{z2 } and p (z2 ) = p(y) + p(z2 ). It follows from Proposition 2.2 (and the fact that C2 is Z3 -connected) that G admits a nowhere-zero map with boundary p . It follows easily from this that G admits a nowhere-zero map with boundary p. We shall require an additional definition and two lemmas before proving Theorems 1.7 and 1.8. Let G, H be graphs, let uv ∈ E(G) and let u v ∈ E(H ). Form a new graph from the disjoint union of G\{uv} and H by identifying u and u and identifying v and v . This operation is called attaching G and H over the edges uv and u v . The graph obtained from this process is said to be a two sum of G and H. Lemma 2.5. Let G1 , G2 be graphs and let H be a two sum of G1 and G2 . (i) If neither G1 nor G2 admits a nowhere-zero Z3 -flow then H does not admit a nowhere-zero Z3 -flow. (ii) If neither G1 nor G2 is Z3 -connected, then H is not Z3 -connected. Proof. Let H be obtained by attaching G1 and G2 over the edges u1 v1 and u2 v2 . Let u, v ∈ V (H ) be the vertices (i) let pi : V (Gi ) → Z3 be given by obtained by identifying u1 and u2 and identifying v1 and v2 , respectively. In case pi (v) = 0 for i = 1, 2. In case (ii) for i = 1, 2 we choose pi : V (Gi ) → Z3 so that v∈V (Gi ) pi (v) = 0 and so that there does not exist a nowhere-zero map : E(Gi ) → Z3 with boundary pi . Define q : V (H ) → Z3 by the following rule: ⎧ p (u ) + p2 (u2 ) if w = u ⎪ ⎨ 1 1 p1 (v1 ) + p2 (v2 ) if w = v q(w) = if w ∈ V (G1 )\{u1 , v1 } ⎪ ⎩ p1 (w) if w ∈ V (G2 )\{u2 , v2 } p2 (w) To resolve cases (i) and (ii) it suffices to prove that there does not exist a nowhere-zero map on E(H ) with boundary q. Suppose (for a contradiction) that there does exist such a map : E(H ) → Z3 . Let i = |E(Gi )\{ui vi } for i = 1, 2. By construction, i is a nowhere-zero map of Gi \ui vi and ji (w) = pi (w) for every w ∈ V (Gi )\{ui , vi }. Since v∈V (Gi ) pi (v) = 0 for each i, and ji (w) = pi (w) for every w ∈ V (Gi )\{ui , vi }, we have that ji (ui ) + ji (vi ) = pi (ui ) + pi (vi ). If ji (ui ) − pi (ui ) = = 0, then let i (ui vi ) = , and we, therefore, obtain a nowhere-zero map i for Gi with ji = pi . This contradicts our assumption that Gi does not admit anowhere-zero map i with ji = pi . So, ji (ui ) = pi (ui ),
j(vi ) = pi (vi )
(1)
for each i = 1, 2. Let be the same as restricted on E(H )\{uv}. That is, = |E(H )\{uv} : E(H )\{uv} → Z3 \{0}. Since (uv) = 0, we have that j(u) = j (u)
and (v) = j (v).
(2)
However, since i = |E(Gi )\{ui vi } for i = 1, 2, and j = q, we have that, by (1), j (u) = j1 (u1 ) + j2 (u2 ) = p1 (u1 ) + p2 (u2 ) = q(u) = j(u). This contradicts (2). This completes the proof.
Lemma 2.6. ¯ 1,2,3 then G2 is not Z3 -connected. (i) If G ∈ T ¯ 1,3 then G2 does not admit a nowhere-zero Z3 -flow. (ii) If G ∈ T ¯ ¯ 1,3 then G2 admits a nowhere-zero Z3 -flow. (iii) If G ∈ T1,2,3 \T
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Proof. We shall prove (i) and (ii) simultaneously by induction on |V (G)|. In both cases, the statement is trivial if |V (G)| 4. For a graph with |V (G)| > 4, we may choose a cut-edge uv ∈ E(G) so that deg(u) > 1 and deg(v) > 1. Let X, Y be the vertex sets of the two components of G\uv and assume that u ∈ X and v ∈ Y . Let G1 = G\(X\{u}) and let G2 = G\(Y \{v}). Now G2 is a two sum of G21 and G22 , so the result follows by Lemma 2.5 and by induction on G1 and G2 . To prove (iii) we will proceed by induction on |V (G)|. The statement is trivial if |V (G)| 4. Suppose that there exist x, y, z ∈ V (G) so that z is adjacent to x and y, deg(z) = 3, and either deg(x) = deg(y) = 1 or xy ∈ E(G) (note that in the latter case deg(x) = deg(y) = 2). It follows by induction that (G\{x, y})2 admits a nowhere-zero Z3 -flow. Since the edges in G2 but not in (G\{x, y})2 form a graph with a nowhere-zero Z3 -flow (they form a graph isomorphic to K4 minus an edge) we find that G admits a Z3 -flow as desired. Thus we may assume that such vertices x, y, z do not exist. Note that this implies in particular that G is a tree. Let v1 , v2 , . . . , vk−1 , vk be the vertex sequence of the longest path of G. It follows from our assumptions that v2 and vk−1 are distinct vertices, deg(v2 ) = deg(vk−1 ) = 2 and deg(v1 ) = 1. Now, by induction G\v1 admits a nowhere-zero Z3 -flow. By altering this flow on the edge v1 v2 it may be extended to a nowhere-zero Z3 -flow of G. This completes the proof.
H1
H2
For positive integers n, m let Kn,m denote the complete bipartite graph on n opposite m vertices. Let H1 be a graph obtained from a circuit of length four by adding a new vertex adjacent to exactly one of the existing vertices. Let H2 be a graph obtained from a circuit of length three by adding two new vertices, each adjacent to a distinct point on the original circuit. The above figure depicts H1 and H2 . Proof of Theorem 1.7. First we shall establish the following claim. ¯ 1,2,3 then G has K1,4 , or H1 , or H2 , or Cn for some n 5 as a subgraph. Claim. If G ∈ /T Proof. We shall prove the contrapositive, so we assume that G does not contain any of the graphs listed above. If G ¯ 1,2,3 . If it has no 4-circuit, then G has maximum degree 3 has a 4-circuit, then it is clear that |V (G)| = 4 so G ∈ T ¯ 1,2,3 . and every circuit of G is a triangle which contains two degree two vertices, so we have that G ∈ T 2 ¯ ¯ 1,2,3 , then If G ∈ T1,2,3 , then it follows from part (i) of Lemma 2.6 that G is not Z3 -connected. If G ∈ /T G contains one of the graphs listed in the above claim as a subgraph. If G contains Cn for some n 5 as a subgraph, then G2 is Z3 -connected by Proposition 2.2 and Corollary 2.3. If G contains K1,4 , H1 , or H2 , then G2 is Z3 connected by Propositions 2.2 and 2.4 and the observation that the square of each of these three graphs contains W4 as a subgraph. Proof of Theorem 1.8. This theorem follows immediately from Theorem 1.7, and parts (ii) and (iii) of Lemma 2.6. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, London, 1976. [2] F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182. [3] M. Kochol, An equivalent version of the 3-flow conjecture, J. Combin. Theory Ser. B 83 (2) (2001) 258–261. [4] H.-J. Lai, Nowhere-zero 3-flows in locally connected graphs, J. Graph Theory 42 (2003) 211–219. [5] R. Xu, C.-Q. Zhang, Nowhere-zero 3-flows in squares of graphs, Electronic J. Combin. 10 (2003) R5.
Further reading [6] C.-Q. Zhang, Integer Flows and Cycle Covers of Graphs, Marcel Dekker, New York, 1997.
Nowhere-zero Z3-flows through Z3-connectivity Matt DeVosa , Rui Xub,∗ , Gexin Yuc a Department of Mathematics, Princeton University, Princeton, NJ 08544, USA b Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA c Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 2 September 2005; received in revised form 14 October 2005; accepted 21 October 2005 Available online 27 December 2005
Abstract Let be an abelian group. Jaeger et al. [Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182] introduced a class of graphs which they call -connected. The main interest in -connected graphs is that every -connected graph admits a nowhere-zero -flow. In this paper, we found some families of Z3 -connected graphs. Our results generalize an early theorem by Lai (Nowhere-zero 3-flows in locally connected graphs, J. Graph Theory 42 (2003) 211–219) for nowhere-zero Z3 -flows in locally connected graphs, and provide a simplified proof of a theorem of Xu and Zhang on nowhere-zero Z3 -flows in squares of graphs. © 2005 Elsevier B.V. All rights reserved. Keywords: Integer flow; Group connectivity; Squares of graphs
1. Introduction Throughout this article, graphs and directed graphs may have multiple edges and loops. Let G be a directed graph, let be an abelian group and let : E(G) → be a map. is nowhere-zero if 0 ∈ / (E(G)). The boundary We say that of is the map j : V (G) → where j(v) = e∈E + (v) (e) − e∈E − (v) (e) for each vertex v ∈ V (G) (note that v∈V (G) j(v) = 0), where E + (v) (or E − (v)) is the set of all arcs of G with their tails (or, heads, respectively) at the vertex v. If j is identically zero, then is called a flow or a -flow. We say that G is -connected if, for every p : V (G) → with v∈V (G) p(v) = 0, there exists a nowhere-zero map : E(G) → with boundary j = p. If we change the orientation of the edge e and switch (e) to −(e), then the boundary is preserved, and the new map is nowhere-zero if and only if the original was. Thus the existence of a nowhere-zero map with a specified boundary depends only on the underlying undirected graph and not on the orientation of the edges. Accordingly, we say that an undirected graph admits a nowhere-zero -flow (is -connected) if some (and thus every) orientation of it admits a nowhere-zero -flow (is -connected). In the remainder of the paper, we will restrict our attention to the group Z3 = Z/3Z. The main interest in nowhere-zero Z3 -flows stems from the following conjecture of Tutte.
∗ Corresponding author. Tel.: +1 678 839 4122; fax: +1 678 839 6490.
E-mail addresses: [email protected] (M. DeVos), [email protected] (R. Xu), [email protected] (G. Yu). 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.10.019
M. DeVos et al. / Discrete Mathematics 306 (2006) 26 – 30
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Conjecture 1.1 (The 3-flow conjecture, Tutte; see unsolved problem 48 in [1]). Every 4-edge-connected graph admits a nowhere-zero Z3 -flow. In their article on group connectivity, Jaeger et al. make the following conjecture on Z3 -connected graphs. Conjecture 1.2 (Jaeger et al. [2]). Every 5-edge connected graph is Z3 -connected. Recently Kochol [3] has shown that Conjecture 1.1 can be reduced to 5-edge-connected graphs. It follows from this that Conjecture 1.2 implies Conjecture 1.1. Thus, it may be possible to achieve the 3-flow conjecture by instead proving a result on Z3 -connectivity. Although we appear to be quite far from a resolution to either of these conjectures, the idea of using Z3 -connectivity to establish results on Z3 -flows is quite useful. Indeed, this is the main theme of this paper. Before stating our main theorems, we shall introduce a definition and an easy observation which is also suggestive of this theme. If H is a connected subgraph of the graph G, then G contracts H, denoted by G/H , is the graph obtained from G by deleting all edges in H and then identifying V (H ) to a single new vertex. The following observation follows easily from the definitions. Observation 1.3. Let H be a Z3 -connected subgraph of the graph G. (i) If G/H admits a nowhere-zero Z3 -flow, then so does G. (ii) If G/H is Z3 -connected, then so is G. Let us call a Z3 -connected graph on > 1 vertex reducible. Now suppose that F is a class of graphs which is closed under contracting subgraphs (for instance the class of all 5-edge-connected graphs). In order to prove that every graph in F admits a nowhere-zero Z3 -flow, it suffices to consider only those graphs which do not contain a reducible subgraph. Similarly, to prove that every graph in F is Z3 -connected, it suffices to show that every graph in F contains a reducible subgraph. This is suggestive of a kind of reducibility/unavoidability approach to Conjecture 1.2. An obvious difficulty in this approach is that for any finite list of reducible graphs H1 , H2 , . . . , Hk , there exist 5-edge-connected graphs containing no subgraph isomorphic to Hi for 1 i k (this follows from the existence of 5-edge-connected graphs of high girth and the fact that every reducible graph contains a circuit). It is possible to prove that graphs with some added structure are Z3 -connected. A graph G is locally connected if G[N(v)] is connected for every v ∈ V (G). We define G to be triangle connected if for every e, f ∈ E(G), there exists a sequence of circuits C1 , C2 , . . . , Ck so that e ∈ E(C1 ), f ∈ E(Ck ), |E(Ci )| 3 for 1 i k, and E(Cj )∩E(Cj +1 ) = ∅ for 1 j k − 1. It is easy to see that every locally connected graph on at least three vertices is also triangle connected. The following theorem gives a sufficient condition for a graph to be Z3 -connected. Theorem 1.4. If G is a loopless triangle connected graph with minimum degree 4 then G is Z3 -connected. This theorem is a generalization of the following result of Lai. Theorem 1.5 (Lai [4]). If G is 2-edge-connected and G[N (v)] is 3-edge-connected for every v ∈ V (G) then G has a nowhere-zero Z3 -flow. To see that Theorem 1.4 implies Theorem 1.5, note that every graph G on > 2 vertices satisfying the input condition of Theorem 1.5 is 2-connected and triangle connected. If v ∈ V (G) and u ∈ N (v), then degG[N(u)] (v) 3, so degG (v) 4. Thus the result follows from Theorem 1.4. If G is a simple and loopless graph, then G2 is the graph with V (G2 ) = V (G) so that u, v ∈ V (G2 ) are adjacent (in 2 G ) if and only if they are distance 2 in G. Using our techniques, we establish necessary and sufficient conditions on G for G2 to be Z3 -connected. As a consequence of this, we obtain a simplified proof of a theorem of Xu and Zhang [5] which gives necessary and sufficient conditions on G for G2 to admit a nowhere-zero Z3 -flow. Before stating this results, we require the following definition.
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Definition 1.6. T1,3 = {T |T is a tree and dT (v) ∈ {1, 3} for every v ∈ V (T )}. ¯ 1,3 = {T |T ∈ T1,3 or T is a simple 4-vertex graph containing a 4-circuit or T is obtained from some T ∈ T1,3 T by adding some edges which join two distance 2 leaves of T }. T1,2,3 = {T |T is a tree and dT (v) ∈ {1, 2, 3} for every v ∈ V (T )}. ¯ 1,2,3 = {T |T ∈ T1,2,3 or T is a simple 4-vertex graph containing a 4-circuit or T is obtained from T some T ∈ T1,2,3 by adding some edges which join two distance 2 leaves of T }. ¯ 1,2,3 . Theorem 1.7. Let G be a simple connected graph. Then G2 is Z3 -connected if and only if G ∈ /T Theorem 1.7 is a generalization of the following result obtained recently by Xu and Zhang. Theorem 1.8 (Xu and Zhang [5]). Let G be a simple connected graph. Then G2 admits a nowhere-zero Z3 -flow if and ¯ 1,3 . only if G ∈ /T 2. Proofs We begin with the following observation which follows easily from the definitions. Observation 2.1. Let H be a Z3 -connected subgraph of the graph G. (i) If V (G) = V (H ), then G is Z3 -connected. (ii) If V (G)\V (H ) = {u} and there are at least two edges joining u to the vertex (or vertices) of V (H ), then G is Z3 -connected. For every integer k 2, let Ck denote a graph which is a circuit on k vertices. A special case of (ii) above is that the graph C2 is Z3 -connected. The proof of Theorem 1.4 requires the following easy proposition. Proposition 2.2. If G is triangle connected and H < G is a Z3 -connected graph on at least two vertices, then G is Z3 -connected. Proof. Choose a maximal Z3 -connected subgraph H of G with H < H . If V (H ) = V (G), then there must exist an edge e ∈ E(H ) and a triangle C < G so that e ∈ E(C) and V (C)V (H ). It now follows from (ii) of Observation 2.1 that H ∪ C is Z3 -connected, contradicting our choice. Proof of Theorem 1.4. If |V (G)| 3 then the theorem is obvious, so we shall assume that |V (G)| 4. Choose a maximal triangle connected subgraph H < G with V (H ) = V (G). Since G is triangle connected, it follows easily that V (G)\V (H ) contains a single vertex, say u. It now follows from our assumptions that we may choose an edge e ∈ E(H ) and a triangle C < G so that u ∈ V (C) and e ∈ E(C). Modify the graph G to form a new graph G by deleting E(C)\{e} and adding a new edge e parallel to e. Let H < G be obtained from H by adding the edge e . If G is Z3 -connected, then it follows easily that G is also Z3 -connected. Thus, the theorem follows from the fact that H is Z3 -connected (by Proposition 2.2 and the fact that C2 is Z3 -connected) and part (ii) of Observation 2.1. Corollary 2.3. If n 5, then Cn2 is Z3 -connected, where Cn is a circuit of length n. Next we will prove that another class of graphs is Z3 -connected. For every n 3, we define Wn to be a simple graph obtained from a circuit of length n by adding a new vertex adjacent to all existing vertices. Proposition 2.4. For every integer k 2, the graph W2k is Z3 -connected. Proof. Let G be a graph obtained from a circuit of length 2k by adding a new vertex x adjacent to all existing vertices. Let p : V (G) → Z3 be a map with v∈V (G) p(v) = 0. If p is identically zero, then it is easy to verify that G
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admits a nowhere-zero map with boundary p. Otherwise, we may choose a vertex y ∈ V (G)\{x} with p(y) = 0. Let N(y) = {x, z1 , z2 }, let G be the graph obtained from G\y by adding a new edge (in parallel) from z1 to x, and let p : V (G ) → Z3 be given by the rule p (v) = p(v) if v ∈ V (G )\{z2 } and p (z2 ) = p(y) + p(z2 ). It follows from Proposition 2.2 (and the fact that C2 is Z3 -connected) that G admits a nowhere-zero map with boundary p . It follows easily from this that G admits a nowhere-zero map with boundary p. We shall require an additional definition and two lemmas before proving Theorems 1.7 and 1.8. Let G, H be graphs, let uv ∈ E(G) and let u v ∈ E(H ). Form a new graph from the disjoint union of G\{uv} and H by identifying u and u and identifying v and v . This operation is called attaching G and H over the edges uv and u v . The graph obtained from this process is said to be a two sum of G and H. Lemma 2.5. Let G1 , G2 be graphs and let H be a two sum of G1 and G2 . (i) If neither G1 nor G2 admits a nowhere-zero Z3 -flow then H does not admit a nowhere-zero Z3 -flow. (ii) If neither G1 nor G2 is Z3 -connected, then H is not Z3 -connected. Proof. Let H be obtained by attaching G1 and G2 over the edges u1 v1 and u2 v2 . Let u, v ∈ V (H ) be the vertices (i) let pi : V (Gi ) → Z3 be given by obtained by identifying u1 and u2 and identifying v1 and v2 , respectively. In case pi (v) = 0 for i = 1, 2. In case (ii) for i = 1, 2 we choose pi : V (Gi ) → Z3 so that v∈V (Gi ) pi (v) = 0 and so that there does not exist a nowhere-zero map : E(Gi ) → Z3 with boundary pi . Define q : V (H ) → Z3 by the following rule: ⎧ p (u ) + p2 (u2 ) if w = u ⎪ ⎨ 1 1 p1 (v1 ) + p2 (v2 ) if w = v q(w) = if w ∈ V (G1 )\{u1 , v1 } ⎪ ⎩ p1 (w) if w ∈ V (G2 )\{u2 , v2 } p2 (w) To resolve cases (i) and (ii) it suffices to prove that there does not exist a nowhere-zero map on E(H ) with boundary q. Suppose (for a contradiction) that there does exist such a map : E(H ) → Z3 . Let i = |E(Gi )\{ui vi } for i = 1, 2. By construction, i is a nowhere-zero map of Gi \ui vi and ji (w) = pi (w) for every w ∈ V (Gi )\{ui , vi }. Since v∈V (Gi ) pi (v) = 0 for each i, and ji (w) = pi (w) for every w ∈ V (Gi )\{ui , vi }, we have that ji (ui ) + ji (vi ) = pi (ui ) + pi (vi ). If ji (ui ) − pi (ui ) = = 0, then let i (ui vi ) = , and we, therefore, obtain a nowhere-zero map i for Gi with ji = pi . This contradicts our assumption that Gi does not admit anowhere-zero map i with ji = pi . So, ji (ui ) = pi (ui ),
j(vi ) = pi (vi )
(1)
for each i = 1, 2. Let be the same as restricted on E(H )\{uv}. That is, = |E(H )\{uv} : E(H )\{uv} → Z3 \{0}. Since (uv) = 0, we have that j(u) = j (u)
and (v) = j (v).
(2)
However, since i = |E(Gi )\{ui vi } for i = 1, 2, and j = q, we have that, by (1), j (u) = j1 (u1 ) + j2 (u2 ) = p1 (u1 ) + p2 (u2 ) = q(u) = j(u). This contradicts (2). This completes the proof.
Lemma 2.6. ¯ 1,2,3 then G2 is not Z3 -connected. (i) If G ∈ T ¯ 1,3 then G2 does not admit a nowhere-zero Z3 -flow. (ii) If G ∈ T ¯ ¯ 1,3 then G2 admits a nowhere-zero Z3 -flow. (iii) If G ∈ T1,2,3 \T
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Proof. We shall prove (i) and (ii) simultaneously by induction on |V (G)|. In both cases, the statement is trivial if |V (G)| 4. For a graph with |V (G)| > 4, we may choose a cut-edge uv ∈ E(G) so that deg(u) > 1 and deg(v) > 1. Let X, Y be the vertex sets of the two components of G\uv and assume that u ∈ X and v ∈ Y . Let G1 = G\(X\{u}) and let G2 = G\(Y \{v}). Now G2 is a two sum of G21 and G22 , so the result follows by Lemma 2.5 and by induction on G1 and G2 . To prove (iii) we will proceed by induction on |V (G)|. The statement is trivial if |V (G)| 4. Suppose that there exist x, y, z ∈ V (G) so that z is adjacent to x and y, deg(z) = 3, and either deg(x) = deg(y) = 1 or xy ∈ E(G) (note that in the latter case deg(x) = deg(y) = 2). It follows by induction that (G\{x, y})2 admits a nowhere-zero Z3 -flow. Since the edges in G2 but not in (G\{x, y})2 form a graph with a nowhere-zero Z3 -flow (they form a graph isomorphic to K4 minus an edge) we find that G admits a Z3 -flow as desired. Thus we may assume that such vertices x, y, z do not exist. Note that this implies in particular that G is a tree. Let v1 , v2 , . . . , vk−1 , vk be the vertex sequence of the longest path of G. It follows from our assumptions that v2 and vk−1 are distinct vertices, deg(v2 ) = deg(vk−1 ) = 2 and deg(v1 ) = 1. Now, by induction G\v1 admits a nowhere-zero Z3 -flow. By altering this flow on the edge v1 v2 it may be extended to a nowhere-zero Z3 -flow of G. This completes the proof.
H1
H2
For positive integers n, m let Kn,m denote the complete bipartite graph on n opposite m vertices. Let H1 be a graph obtained from a circuit of length four by adding a new vertex adjacent to exactly one of the existing vertices. Let H2 be a graph obtained from a circuit of length three by adding two new vertices, each adjacent to a distinct point on the original circuit. The above figure depicts H1 and H2 . Proof of Theorem 1.7. First we shall establish the following claim. ¯ 1,2,3 then G has K1,4 , or H1 , or H2 , or Cn for some n 5 as a subgraph. Claim. If G ∈ /T Proof. We shall prove the contrapositive, so we assume that G does not contain any of the graphs listed above. If G ¯ 1,2,3 . If it has no 4-circuit, then G has maximum degree 3 has a 4-circuit, then it is clear that |V (G)| = 4 so G ∈ T ¯ 1,2,3 . and every circuit of G is a triangle which contains two degree two vertices, so we have that G ∈ T 2 ¯ ¯ 1,2,3 , then If G ∈ T1,2,3 , then it follows from part (i) of Lemma 2.6 that G is not Z3 -connected. If G ∈ /T G contains one of the graphs listed in the above claim as a subgraph. If G contains Cn for some n 5 as a subgraph, then G2 is Z3 -connected by Proposition 2.2 and Corollary 2.3. If G contains K1,4 , H1 , or H2 , then G2 is Z3 connected by Propositions 2.2 and 2.4 and the observation that the square of each of these three graphs contains W4 as a subgraph. Proof of Theorem 1.8. This theorem follows immediately from Theorem 1.7, and parts (ii) and (iii) of Lemma 2.6. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, London, 1976. [2] F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182. [3] M. Kochol, An equivalent version of the 3-flow conjecture, J. Combin. Theory Ser. B 83 (2) (2001) 258–261. [4] H.-J. Lai, Nowhere-zero 3-flows in locally connected graphs, J. Graph Theory 42 (2003) 211–219. [5] R. Xu, C.-Q. Zhang, Nowhere-zero 3-flows in squares of graphs, Electronic J. Combin. 10 (2003) R5.
Further reading [6] C.-Q. Zhang, Integer Flows and Cycle Covers of Graphs, Marcel Dekker, New York, 1997.