Group connectivity in line graphs

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)?A with ∑_v∈V(G)b(v)=0, there always exists a map f:E(G)?A−{0}, such that at each v∈V(G), in A, then G is A-connected. Let Z₃ denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z₃-connected. In this paper, we proved the following.

• (i) 

  Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected.

• (ii) 

  Every 6-edge-connected triangular line graph is Z3-connected.

• (iii) 

  Every 7-edge-connected triangular claw-free graph is Z3-connected.

In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.     

On Connected [g,f +1]-Factors in Graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z⁺-{1} such that g(v) ≤ f(v) ≤ d_G(v) for all v ∈V (G), where d_G(v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs. * The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025. † The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).     

Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non‐negative integer. We say G is k‐antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted‐k‐antimagic if for any vertex weight function w: V→?, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well‐known conjecture asserts that every connected graph G≠K₂ is 0‐antimagic. On the other hand, there are connected graphs G≠K₂ which are not weighted‐1‐antimagic. It is unknown whether every connected graph G≠K₂ is weighted‐2‐antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted‐2‐antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted‐1‐antimagic. We also prove that every connected graph G≠K₂ on n vertices is weighted‐ ?3n/2?‐antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

Group Chromatic Number of Graphs without K5-Minors

 Let G be a graph with a fixed orientation and let A be a group. Let F(G,A) denote the set of all functions f: E(G) ↦A. The graph G is A -colorable if for any function f∈F(G,A), there is a function c: V(G) ↦A such that for every directed e=u v∈E(G), c(u)−c(v)≠f(e). The group chromatic numberχ₁(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m under a given orientation D. In [J. Combin. Theory Ser. B, 56 (1992), 165–182], Jaeger et al. proved that if G is a simple planar graph, then χ₁(G)≤6. We prove in this paper that if G is a simple graph without a K ₅-minor, then χ₁(G)≤5. Received: August 18, 1999 Final version received: December 12, 2000     

Balloons,cut‐edges,matchings, and total domination in regular graphs of odd degree

A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}$ be the family of connected (2r+1)‐regular graphs with n vertices, and let ${{b}}={{max}}\{{{b}}({{G}}): {{G}}\in {\mathcal{F}}_{{{n}},{{r}}}\}$. For ${{G}}\in{\mathcal{F}}_{{{n}},{{r}}}$, we prove the sharp inequalities c(G)?[r(n?2)?2]/(2r²+2r?1)?1 and α′(G)?n/2?rb/(2r+1). Using b?[(2r?1)n+2]/(4r²+4r?2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ_t(G) of a cubic graph, we prove γ_t(G)?n/2?b(G)/2 (except that γ_t(G) may be n/2?1 when b(G)=3 and the balloons cover all but one vertex). With α′(G)?n/2?b(G)/3 for cubic graphs, this improves the known inequality γ_t(G)?α′(G). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 116–131, 2010     

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χ_g(G). In this note we will prove the following results. (1) Let H₁ and H₂ be two subgraphs of G such that V(H₁)∩V(H₂)=∅ and V(H₁)∪V(H₂)=V(G). Then χ_g(G)≤min{max{χ_g(H₁), max_v∈V(H2)deg(v,G)+1},max{χ_g(H₂), max_u∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K_3,3-minor, then χ_g(G)≤5.     

Fan‐type theorem for path‐connectivity

For a connected noncomplete graph G, let μ(G):=min{max {d_G(u), d_G(v)}:d_G(u, v)=2}. A well‐known theorem of Fan says that every 2‐connected noncomplete graph has a cycle of length at least min{|V(G)|, 2μ(G)}. In this paper, we prove the following Fan‐type theorem: if G is a 3‐connected noncomplete graph, then each pair of distinct vertices of G is joined by a path of length at least min{|V(G)|?1, 2μ(G)?2}. As consequences, we have: (i) if G is a 3‐connected noncomplete graph with , then G is Hamilton‐connected; (ii) if G is a (s+2)‐connected noncomplete graph, where s≥1 is an integer, then through each path of length s of G there passes a cycle of length≥min{|V(G)|, 2μ(G)?s}. Several results known before are generalized and a conjecture of Enomoto, Hirohata, and Ota is proved. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 265–282, 2002 DOI 10.1002/jgt.10028     

Circular consecutive choosability of k‐choosable graphs

Let S(r) denote a circle of circumference r. The circular consecutive choosability ch_cc(G) of a graph G is the least real number t such that for any r≥χ_c(G), if each vertex v is assigned a closed interval L(v) of length t on S(r), then there is a circular r‐coloring f of G such that f(v)∈L(v). We investigate, for a graph, the relations between its circular consecutive choosability and choosability. It is proved that for any positive integer k, if a graph G is k‐choosable, then ch_cc(G)?k + 1 ? 1/k; moreover, the bound is sharp for k≥3. For k = 2, it is proved that if G is 2‐choosable then ch_cc(G)?2, while the equality holds if and only if G contains a cycle. In addition, we prove that there exist circular consecutive 2‐choosable graphs which are not 2‐choosable. In particular, it is shown that ch_cc(G) = 2 holds for all cycles and for K_{2, n} with n≥2. On the other hand, we prove that ch_cc(G)>2 holds for many generalized theta graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 178‐197, 2011     

List Colorings of K5‐Minor‐Free Graphs With Special List Assignments

The following question was raised by Bruce Richter. Let G be a planar, 3‐connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K₅‐minor‐free r‐connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[S_k], where S_k: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(S_k) in G between the components of G[S_k]. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:18–30, 2012     

Graphic Sequences Have Realizations Containing Bisections of Large Degree

A bisection of a graph is a balanced bipartite spanning sub‐graph. Bollobás and Scott conjectured that every graph G has a bisection H such that deg_H(v) ≥ ?deg_G(v)/2? for all vertices v. We prove a degree sequence version of this conjecture: given a graphic sequence π, we show that π has a realization G containing a bisection H where deg_H(v) ≥ ?(deg_G(v) ? 1)/2? for all vertices v. This bound is very close to best possible. We use this result to provide evidence for a conjecture of Brualdi (Colloq. Int. CNRS, vol. 260, CNRS, Paris) and Busch et al. (2011), that if π and π ? k are graphic sequences, then π has a realization containing k edge‐disjoint 1‐factors. We show that if the minimum entry δ in π is at least n/2 + 2, then π has a realization containing edge‐disjoint 1‐factors. We also give a construction showing the limits of our approach in proving this conjecture. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Lower negative decision number in a graph

Let G=(V,E) be a graph. A function f:V(G)→{?1,1} is called bad if ∑ _v∈N(v) f(v)≤1 for every v∈V(G). A bad function f of a graph G is maximal if there exists no bad function g such that g≠f and g(v)≥f(v) for every v∈V. The minimum of the values of ∑ _v∈V f(v), taken over all maximal bad functions f, is called the lower negative decision number and is denoted by β _D ^* (G). In this paper, we present sharp lower bounds on this number for regular graphs and nearly regular graphs, and we also characterize the graphs attaining those bounds.     

Nowhere‐zero 3‐flows in locally connected graphs

Let G be a graph. For each vertex v ∈V(G), N_v denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k‐edge‐connected if for each vertex v ∈V(G), N_v is k‐edge‐connected. In this paper we study the existence of nowhere‐zero 3‐flows in locally k‐edge‐connected graphs. In particular, we show that every 2‐edge‐connected, locally 3‐edge‐connected graph admits a nowhere‐zero 3‐flow. This result is best possible in the sense that there exists an infinite family of 2‐edge‐connected, locally 2‐edge‐connected graphs each of which does not have a 3‐NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211–219, 2003     

On Meyniel's conjecture of the cop number

Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper‐bound O(n/ lnn) due to Chiniforooshan.     

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum‐size matching and it is factor critical if has a perfect matching for each vertex v of G. It is known that any 2‐connected equimatchable graph is either bipartite or factor critical. We prove that for 2‐connected factor‐critical equimatchable graph G the graph is either or for some n for any vertex v of G and any minimal matching M such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2‐connected equimatchable factor‐critical graphs embeddable in the orientable surface of genus g to if and to if . Moreover, for any nonnegative integer g we construct a 2‐connected equimatchable factor‐critical graph with genus g and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2‐connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face‐width k. Finally, we prove that any d‐degenerate 2‐connected equimatchable factor‐critical graph has at most vertices, where a graph is d‐degenerate if every its induced subgraph contains a vertex of degree at most d.     

2‐ and 3‐factors of graphs on surfaces

It has been conjectured that any 5‐connected graph embedded in a surface Σ with sufficiently large face‐width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4‐connected graphs. In this article, we shall study the existence of 2‐ and 3‐factors in a graph embedded in a surface Σ. A hamiltonian cycle is a special case of a 2‐factor. Thus, it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2‐factor. In fact, we only need the 4‐connectivity with minimum degree at least 5. In addition, our face‐width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface Σ of Euler genus g.

• (1) If G is 4‐connected and minimum degree of G is at least 5, and furthermore, face‐width of G is at least 4g?12, then G has a 2‐factor.
• (2) If G is 5‐connected and face‐width of G is at least max{44g?117, 5}, then G has a 3‐factor.

The connectivity condition for both results are best possible. In addition, the face‐width conditions are necessary too. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 67:306‐315, 2011     

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

Let G be a simple undirected connected graph on n vertices with maximum degree Δ. Brooks' Theorem states that G has a proper Δ‐coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k‐coloring, , a Δ‐coloring of G can be obtained by a sequence of recolorings using only the original k colors unless

• – G is a complete graph or a cycle with an odd number of vertices, or
• – , G is Δ‐regular and, for each vertex v in G, no two neighbors of v are colored alike.

We use this result to study the reconfiguration graph of the k‐colorings of G. The vertex set of is the set of all possible k‐colorings of G and two colorings are adjacent if they differ on exactly one vertex. We prove that for , consists of isolated vertices and at most one further component that has diameter . This result enables us to complete both a structural and an algorithmic characterization for reconfigurations of colorings of graphs of bounded maximum degree.     

Graphs Containing Topological H

Let H denote the tree with six vertices, two of which are adjacent and of degree 3. Let G be a graph and be distinct vertices of G. We characterize those G that contain a topological H in which are of degree 3 and are of degree 1, which include all 5‐connected graphs. This work was motivated by the Kelmans–Seymour conjecture that 5‐connected nonplanar graphs contain topological K₅.     

Upper signed domination number

Let G=(V,E) be a graph. A signed dominating function on G is a function f:V→{-1,1} such that for each v∈V, where N[v] is the closed neighborhood of v. The weight of a signed dominating function f is . A signed dominating function f is minimal if there exists no signed dominating function g such that g≠f and g(v)?f(v) for each v∈V. The upper signed domination number of a graph G, denoted by Γ_s(G), equals the maximum weight of a minimal signed dominating function of G. In this paper, we establish an tight upper bound for Γ_s(G) in terms of minimum degree and maximum degree. Our result is a generalization of those for regular graphs and nearly regular graphs obtained in [O. Favaron, Signed domination in regular graphs, Discrete Math. 158 (1996) 287-293] and [C.X. Wang, J.Z. Mao, Some more remarks on domination in cubic graphs, Discrete Math. 237 (2001) 193-197], respectively.     

Group connectivity and group colorings of graphs — A survey

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A − {0}. The graph G is A-connected if G has an orientation D(G) such that for every map b: V (G) ↦ A satisfying Σ _v∈V(G) b(v) = 0, there is a function f: E(G) ↦ A* such that for each vertex v ∈ V (G), the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b(v). The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections.

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).