Hamiltonicity and forbidden subgraphs in 4‐connected graphs

Let T be the line graph of the unique tree F on 8 vertices with degree sequence (3,3,3,1,1,1,1,1), i.e., T is a chain of three triangles. We show that every 4‐connected {T, K_1,3}‐free graph has a hamiltonian cycle. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 262–272, 2005     

Path-Factors in the Square of a Tree

In this paper, we give a necessary condition for the existence of P ₃-factor in the square of a tree. Furthermore, we show that the square of any tree on at least 6 vertices has a {P ₃,P ₄}-factor. As a consequence, every connected graph on at least 6 vertices has a {P ₃,P ₄}-factor. Zhao Zhang: The research is supported by NSFC (60603003) and XJEDU. Received: November 15, 2006. Final version received: December 12, 2007.     

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     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

K4−‐factor in a graph

Let G be a graph of order 4k and let δ(G) denote the minimum degree of G. Let F be a given connected graph. Suppose that |V(G)| is a multiple of |V(F)|. A spanning subgraph of G is called an F‐factor if its components are all isomorphic to F. In this paper, we prove that if δ(G)≥5/2k, then G contains a K₄^?‐factor (K₄^? is the graph obtained from K₄ by deleting just one edge). The condition on the minimum degree is best possible in a sense. In addition, the proof can be made algorithmic. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 111–128, 2002     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000     

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K ₂, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each k 3 2, q 3 \binomk+12{k\geq 2,\,q\geq\binom{k+1}{2}} with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.     

Goal-minimally <Emphasis Type="Italic">k</Emphasis>-elongated graphs

Let k be an integer. A 2-edge connected graph G is said to be goal-minimally k-elongated (k-GME) if for every edge uv ∈ E(G) the inequality d _G−uv (x, y) > k holds if and only if {u, v} = {x, y}. In particular, if the integer k is equal to the diameter of graph G, we get the goal-minimally k-diametric (k-GMD) graphs. In this paper we construct some infinite families of GME graphs and explore k-GME and k-GMD properties of cages. This research was supported by the Slovak Scientific Grant Agency VEGA No. 1/0406/09.     

Orthogonal factorizations of graphs

Let G be a graph with vertex set V(G) and edge set E(G). Let k₁, k₂,…,k_m be positive integers. It is proved in this study that every [0,k₁+…+k_m?m+1]‐graph G has a [0, k_i]₁^m‐factorization orthogonal to any given subgraph H with m edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 267–276, 2002     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

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     

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     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

Group connectivity of complementary graphs

Let G be a 2‐edge‐connected undirected graph, A be an (additive) abelian group and A* = A?{0}. A graph G is A‐connected if G has an orientation D(G) such that for every function b: V(G)?A satisfying , 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 equals b(v). For a 2‐edge‐connected graph G, define Λ_g(G) = min{k: for any abelian group A with |A|?k, G is A‐connected }. In this article, we prove the following Ramsey type results on group connectivity:

1. Let G be a simple graph on n?6 vertices. If min{δ(G), δ(G^c)}?2, then either Λ_g(G)?4, or Λ_g(G^c)?4.
2. Let Z₃ denote the cyclic group of order 3, and G be a simple graph on n?44 vertices. If min{δ(G), δ(G^c)}?4, then either G is Z₃‐connected, or G^c is Z₃‐connected. © 2011 Wiley Periodicals, Inc. J Graph Theory

     

An eulerian trail traversing specified edges in given order

The following results are proved in this paper. Let G be a 2k-edge-connected eulerian graph. (i) For every set {e₁, e₂, ?, e_2k+1} ? E(G) there is an eulerian trail T of the form e₁, e₂, ?, e_2k+1, ?. (ii) For every set E* = {e₁, e₂, ?, e_k} ? E(G) there is an eulerian trail T = e₁, ?, e₂, ?, e_k, ? in which the elements of E* are traversed in accordance with a prescribed orientation. © 1995 John Wiley & Sons, Inc.     

An Anti-Ramsey Theorem on Diamonds

Let G be the diamond (the graph obtained from K ₄ by deleting an edge) and, for every n ≥ 4, let f(n, G) be the minimum integer k such that, for every edge-coloring of the complete graph of order n which uses exactly k colors, there is at least one copy of G all whose edges have different colors. Let ext(n, {C ₃, C ₄}) be the maximum number of edges of a graph on n vertices free of triangles and squares. Here we prove that for every n ≥ 4,

Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Claw‐decompositions and tutte‐orientations

We conjecture that, for each tree T, there exists a natural number k_T such that the following holds: If G is a k_T‐edge‐connected graph such that |E(T)| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T. We prove that for T = K_1,3 (the claw), this holds if and only if there exists a (smallest) natural number k_t such that every k_t‐edge‐connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3‐flow conjecture says that k_t = 4. We prove the weaker statement that every 4$\lceil$ log n$\rceil$ ‐edge‐connected graph with n vertices has an edge‐decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge‐decomposition into claws. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 135–146, 2006     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.