List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

Facial non‐repetitive edge‐coloring of plane graphs

A sequence r₁, r₂, …, r_2n such that r_i=r_{n+ i} for all 1≤i≤n is called a repetition. A sequence S is called non‐repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non‐repetitive if the sequence of colors of its edges is non‐repetitive. If G is a plane graph, a facial non‐repetitive edge‐coloring of G is an edge‐coloring such that any facial trail (i.e. a trail of consecutive edges on the boundary walk of a face) is non‐repetitive. We denote π′_f(G) the minimum number of colors of a facial non‐repetitive edge‐coloring of G. In this article, we show that π′_f(G)≤8 for any plane graph G. We also get better upper bounds for π′_f(G) in the cases when G is a tree, a plane triangulation, a simple 3‐connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 38–48, 2010     

Crossing properties of graph reliability functions

Let G be a graph and p ϵ (0, 1). Let A(G, p) denote the probability that if each edge of G is selected at random with probability p then the resulting spanning subgraph of G is connected. Then A(G, p) is a polynomial in p. We prove that for every integer k ≥ 1 and every k‐tuple (m₁, m₂, … ,m_k) of positive integers there exist infinitely many pairs of graphs G₁ and G₂ of the same size such that the polynomial A(G₁, p) − A(G₂, p) has exactly k roots x₁ < x₂ < ··· < x_k in (0, 1) such that the multiplicity of x_i is m_i. We also prove the same result for the two‐terminal reliability polynomial, defined as the probability that the random subgraph as above includes a path connecting two specified vertices. These results are based on so‐called A‐ and T‐multiplying constructions that are interesting in themselves. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 206–221, 2000     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

On a conjecture by Plummer and Toft

The cyclic chromatic number χ_c(G) of a 2‐connected plane graph G is the minimum number of colors in an assigment of colors to the vertices of G such that, for every face‐bounding cycle f of G, the vertices of f have different colors. Plummer and Toft proved that, for a 3‐connected plane graph G, under the assumption Δ*(G) ≥ 42, where Δ*(G) is the size of a largest face of G, it holds that χ_c(G) ≤ Δ*(G) + 4. They conjectured that, if G is a 3‐connected plane graph, then χ_c>(G) ≤ Δ*(G) + 2. In the article the conjecture is proved for Δ*(G) ≥ 24. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 177–189, 1999     

On stability of the hamiltonian index under contractions and closures

The hamiltonian index of a graph G is the smallest integer k such that the k‐th iterated line graph of G is hamiltonian. We first show that, with one exceptional case, adding an edge to a graph cannot increase its hamiltonian index. We use this result to prove that neither the contraction of an A_G(F)‐contractible subgraph F of a graph G nor the closure operation performed on G (if G is claw‐free) affects the value of the hamiltonian index of a graph G. AMS Subject Classification (2000): 05C45, 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Acyclic edge coloring of graphs with maximum degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a′(G)?Δ + 2, where Δ=Δ(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Δ(G)?4, with the additional restriction that m?2n?1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m?2n, when Δ(G)?4. It follows that for any graph G if Δ(G)?4, then a′(G)?7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 192–209, 2009     

Asymptotics of the list‐chromatic index for multigraphs

The list‐chromatic index, χ_l′(G) of a multigraph G is the least t such that if S(A) is a set of size t for each A∈E≔E(G), then there exists a proper coloring σ of G with σ(A)∈S(A) for each A∈E. The list‐chromatic index is bounded below by the ordinary chromatic index, χ′(G), which in turn is at least the fractional chromatic index, χ′*(G). In previous work we showed that the chromatic and fractional chromatic indices are asymptotically the same; here we extend this to the list‐chromatic index: χ_l′(G)∼χ′*(G) as χ_l′(G)→∞. The proof uses sampling from “hard‐core” distributions on the set of matchings of a multigraph to go from fractional to list colorings. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 117–156, 2000     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K_t, we have c₁mt²/ln t ≤ f(m, K_t) ≤ c₂mt², and cmt³/ln t ≤ g(m, K_t) ≤ cmt³/ln t, where c₁, c₂, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

Acyclic edge colorings of graphs

A proper coloring of the edges of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

Contractible subgraphs in k‐connected graphs

For a graph G we define a graph T(G) whose vertices are the triangles in G and two vertices of T(G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k‐connected graph and T(G) contains no edge, then G admits a k‐contractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashi's result by showing that if G is k‐connected and the maximum degree of T(G) is at most 1, then G admits a k‐contractible clique of size at most 3 or there exist independent edges e and f of G such that e and f are contained in triangles sharing an edge and G/e/f is k‐connected. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 121–136, 2007     

Graphs omitting sums of complete graphs

For every finite m and n there is a finite set {G₁, …, G_l} of countable (m · K_n)-free graphs such that every countable (m · K_n)-free graph occurs as an induced subgraph of one of the graphs G_l © 1997 John Wiley & Sons, Inc.     

The symmetric (2k,k)‐graphs

A noncomplete graph G is called an (n, k)‐graph if it is n‐connected and G − X is not (n − |X| + 1)‐connected for any X ⊆ V(G) with |X| ≤ k. Mader conjectured that for k ≥ 3 the graph K_{2k + 2} − (1‐factor) is the unique (2k, k)‐graph. We settle this conjecture for strongly regular graphs, for edge transitive graphs, and for vertex transitive graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 35–51, 2001     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

CO‐irredundant Ramsey numbers for graphs

The study of the CO‐irredundant Ramsey numbers t(n₁, ···, n_k) is initiated. It is shown that several values and bounds for these numbers may be obtained from the well‐studied generalized graph Ramsey numbers and the values of t(4, 5), t(4, 6) and t(3, 3, m) are calculated. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 258–268, 2000     

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     

Coloring graphs from lists with bounded size of their union

A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 2^4k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory