Rainbows in the Hypercube

Let Q_n be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples differ in exactly one position. An edge coloring of a graph H is called rainbow if no two edges of H have the same color. Let f(G,H) be the largest number of colors such that there exists an edge coloring of G with f(G,H) colors such that no subgraph isomorphic to H is rainbow. In this paper we start the investigation of this anti-Ramsey problem by providing bounds on f(Q_n,Q_k) which are asymptotically tight for k = 2 and by giving some exact results.     

Gallai–Ramsey numbers for cycles

For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of K_n, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K₃ and H is a cycle of a fixed length.     

On totally multicolored stars

Given an edge‐coloring of a graph G, a subgraph M of G will be called totally multicolored if no two edges of M receive the same color. Let h(G, K_1,q) be the minimum integer such that every edge‐coloring of G using exactly h(G, K_1,q) colors produces at least one totally multicolored copy of K_1,q (the q‐star) in G. In this article, an upper bound of h(G, K_1,q) is presented, as well as some applications of this upper bound. © 2005 Wiley Periodicals, Inc.     

On the perfect orderability of unions of two graphs

A graph G is perfectly orderable, if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph, if it contains no P₄, 2K₂, and C₄ as induced subgraph. A theorem of Chvátal, Hoàng, Mahadev, and de Werra states that a graph is perfectly orderable, if it is the union of two threshold graphs. In this article, we investigate possible generalizations of the above theorem. Hoàng has conjectured that, if G is the union of two graphs G₁ and G₂, then G is perfectly orderable whenever G₁ and G₂ are both P₄‐free and 2K₂‐free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter‐example to this conjecture, and we prove a special case of it: if G₁ and G₂ are two edge‐disjoint graphs that are P₄‐free and 2K₂‐free, then the union of G₁ and G₂ is perfectly orderable. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 32–43, 2000     

Stability,domination and irredundance in a graph

In a graph G, a set X is called a stable set if any two vertices of X are nonadjacent. A set X is called a dominating set if every vertex of V – X is joined to at least one vertex of X. A set X is called an irredundant set if every vertex of X, not isolated in X, has at least one proper neighbor, that is a vertex of V – X joined to it but to no other vertex of X. Let α′ and α, γ, and Γ, ir and IR, denote respectively the minimum and maximum cardinalities of a maximal stable set, a minimal dominating set, and a maximal irredundant set. It is known that ir ? γ ? α′ ? α ? Γ ? IR and that if G does not contain any induced subgraph isomorphic to K_1,3, then γ = α′. Here we prove that if G contains no induced subgraph isomorphic to K_1,3 or to the graph H of figure 1, then ir = γ = α′. We prove also that if G contains no induced subgraph isomorphic to K_1,3, to H, or to the graph h of figure 3, then Γ = IR. Finally, we improve a result of Bollobas and Cockayne about sufficient conditions for γ = ir in terms of forbidden subgraphs.     

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     

Relative colorings of graphs

In this paper, the notion of relative chromatic number χ(G, H) for a pair of graphs G, H, with H a full subgraph of G, is formulated; namely, χ(G, H) is the minimum number of new colors needed to extend any coloring of H to a coloring of G. It is shown that the four color conjecture (4CC) is equivalent to the conjecture (R4CC) that χ(G, H) ≤ 4 for any (possibly empty) full subgraph H of a planar graph G and also to the conjecture (CR3CC) that χ(G, H) ≤ 3 if H is a connected and nonempty full subgraph of planar G. Finally, relative coloring theorems on surfaces other than the plane or sphere are proved.     

Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

An interval coloring of a graph is a proper edge coloring such that the set of used colors at every vertex is an interval of integers. Generally, it is an NP‐hard problem to decide whether a graph has an interval coloring or not. A bipartite graph G = (A,B;E) is (α, β)‐biregular if each vertex in A has degree α and each vertex in B has degree β. In this paper we prove that if the (3,4)‐biregular graph G has a cubic subgraph covering the set B then G has an interval coloring. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 122–128, 2004     

The Ramsey property for graphs with forbidden complete subgraphs

The following theorem is proved: Let G be a finite graph with cl(G) = m, where cl(G) is the maximum size of a clique in G. Then for any integer r ≥ 1, there is a finite graph H, also with cl(H) = m, such that if the edges of H are r-colored in any way, then H contains an induced subgraph G′ isomorphic to G with all its edges the same color.     

Subdivision Extendibility

Let H be a multigraph and G a graph containing a subgraph isomorphic to a subdivision of H, with S ⊂ V(G) (the ground set) the image of V(H) under the isomorphism. We consider connectivity and minimum degree or degree sum conditions sufficient to imply there is a spanning subgraph of G isomorphic to a subdivision of H on the same ground set S. These results generalize a number of theorems in the literature.     

Rainbow faces in edge‐colored plane graphs

A face of an edge‐colored plane graph is called rainbow if the number of colors used on its edges is equal to its size. The maximum number of colors used in an edge coloring of a connected plane graph Gwith no rainbow face is called the edge‐rainbowness of G. In this paper we prove that the edge‐rainbowness of Gequals the maximum number of edges of a connected bridge face factor H of G, where a bridge face factor H of a plane graph Gis a spanning subgraph H of Gin which every face is incident with a bridge and the interior of any one face f∈F(G) is a subset of the interior of some face f′∈F(H). We also show upper and lower bounds on the edge‐rainbowness of graphs based on edge connectivity, girth of the dual graphs, and other basic graph invariants. Moreover, we present infinite classes of graphs where these equalities are attained. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 84–99, 2009     

On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by x′ _Aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.     

Edge‐partitions of planar graphs and their game coloring numbers

Let G be a planar graph and let g(G) and Δ(G) be its girth and maximum degree, respectively. We show that G has an edge‐partition into a forest and a subgraph H so that (i) Δ(H) ≤ 4 if g(G) ≥ 5; (ii) Δ(H) ≤ 2 if g(G) ≥ 7; (iii) Δ(H)≤ 1 if g(G) ≥ 11; (iv) Δ(H) ≤ 7 if G does not contain 4‐cycles (though it may contain 3‐cycles). These results are applied to find the following upper bounds for the game coloring number col_g(G) of a planar graph G: (i) col_g(G) ≤ 8 if g(G) ≥ 5; (ii) col_g(G)≤ 6 if g(G) ≥ 7; (iii) col_g(G) ≤ 5 if g(G) ≥ 11; (iv) col_g(G) ≤ 11 if G does not contain 4‐cycles (though it may contain 3‐cycles). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 307–317, 2002     

Every 3-connected,locally connected,claw-free graph is Hamilton-connected

A graph G is locally connected if the subgraph induced by the neighbourhood of each vertex is connected. We prove that a locally connected graph G of order p ≥ 4, containing no induced subgraph isomorphic to K_1,3, is Hamilton-connected if and only if G is 3-connected. © 1996 John Wiley & Sons, Inc.     

An upper bound for the adjacent vertex distinguishing acyclic edge chromatic number of a graph

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′ _aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′ _aa (G) ≤ 32Δ. Supported by the Natural Science Foundation of Gansu Province (3ZS051-A25-025)     

The minimum degree of Ramsey‐minimal graphs

We write H → G if every 2‐coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G‐minimal if H → G, but for every proper subgraph H′ of H, H′ ? G. We define s(G) to be the minimum s such that there exists a G‐minimal graph with a vertex of degree s. We prove that s(K_k) = (k ? 1)² and s(K_a,b) = 2 min(a,b) ? 1. We also pose several related open problems. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 167–177, 2007     

Subgraphs and the Laplacian spectrum of a graph

Let G be a graph and H a subgraph of G. In this paper, a set of pairwise independent subgraphs that are all isomorphic copies of H is called an H-matching. Denoting by ν(H,G) the cardinality of a maximum H-matching in G, we investigate some relations between ν(H,G) and the Laplacian spectrum of G.     

Circular chromatic index of graphs of maximum degree 3

This paper proves that if G is a graph (parallel edges allowed) of maximum degree 3, then χ′_c(G) ≤ 11/3 provided that G does not contain H₁ or H₂ as a subgraph, where H₁ and H₂ are obtained by subdividing one edge of K (the graph with three parallel edges between two vertices) and K₄, respectively. As χ′_c(H₁) = χ′_c(H₂) = 4, our result implies that there is no graph G with 11/3 < χ′_c(G) < 4. It also implies that if G is a 2‐edge connected cubic graph, then χ′_c(G) ≤ 11/3. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 325–335, 2005     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

Set vertex colorings and joins of graphs

For a nontrivial connected graph G, let c: V (G) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) ≠ NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number x _s (G). A study is made of the set chromatic number of the join G+H of two graphs G and H. Sharp lower and upper bounds are established for x _s (G + H) in terms of x _s (G), x _s (H), and the clique numbers ω(G) and ω(H).