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     

Optimal acyclic edge‐coloring of cubic graphs

An acyclic edge‐coloring of a graph is a proper edge‐coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge‐coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K₄ or K_3,3; the acyclic chromatic index of K₄ and K_3,3 is 5. This result has previously been published by Fiam?ík, but his published proof was erroneous.     

Interval Non‐edge‐Colorable Bipartite Graphs and Multigraphs

An edge‐coloring of a graph G with colors is called an interval t‐coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991, Erd?s constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erd?s's counterexample is the smallest (in a sense of maximum degree) known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non‐edge‐colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.     

The edge‐bandwidth of theta graphs

An edge‐labeling f of a graph G is an injection from E(G) to the set of integers. The edge‐bandwidth of G is B′(G) = min_f {B′(f)} where B′(f) is the maximum difference between labels of incident edges of G. The theta graph Θ(l₁,…,l_m) is the graph consisting of m pairwise internally disjoint paths with common endpoints and lengths l₁ ≤ ··· ≤ l_m. We determine the edge‐bandwidth of all theta graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 89–98, 2000     

Game chromatic index of k‐degenerate graphs

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving first, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when Alice wins, all edges of G are properly colored. The game chromatic index of a graph G is the minimum number of colors for which Alice has a winning strategy. In this paper, we study the edge coloring game on k‐degenerate graphs. We prove that the game chromatic index of a k‐degenerate graph is at most Δ + 3k − 1, where Δ is the maximum vertex degree of the graph. We also show that the game chromatic index of a forest of maximum degree 3 is at most 4 when the forest contains an odd number of edges. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 144–155, 2001     

On Neighbor‐Distinguishing Index of Planar Graphs

A proper edge coloring of a graph G without isolated edges is neighbor‐distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor‐distinguishing index of G is the minimum number ndi(G) of colors in a neighbor‐distinguishing edge coloring of G. Zhang, Liu, and Wang in 2002 conjectured that if G is a connected graph of order at least 6. In this article, the conjecture is verified for planar graphs with maximum degree at least 12.     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

Retracts of box products with odd‐angulated factors

Let G be a connected graph with odd girth 2κ+1. Then G is a (2κ+1)‐angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2κ+1)‐cycle. We prove that if G is (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+3, then any retract of the box (or Cartesian) product G□ H is S□T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2κ+1)‐angulated if any two vertices of G are connected by a sequence of (2κ+1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+1, then any retract of G□ H is S□T where S is a retract of G and T is a connected subgraph of H or |V(S)|=1 and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival [Disc Math 70 ( 13 ), 169–184]. As a corollary, we get that the core of the box product of two strongly (2κ+1)‐angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2κ+1)‐angulated core, then either Gⁿ is a core for all positive integers n, or the core of Gⁿ is G for all positive integers n. In the latter case, G is homomorphically equivalent to a normal Cayley graph [Larose, Laviolette, Tardiff, European J Combin 19 ( 12 ), 867–881]. In particular, let G be a strongly (2κ+1)‐angulated core such that either G is not vertex‐transitive, or G is vertex‐transitive and any two maximum independent sets have non‐empty intersection. Then Gⁿ is a core for any positive integer n. On the other hand, let G_i be a (2κ_i+1)‐angulated core for 1 ≤ i≤ n where κ₁ < κ₂ < … < κ_n. If G_i has a vertex that is fixed under any automorphism for 1 ≤ i ≤ n‐1, or G_i is vertex‐transitive such that any two maximum independent sets have non‐empty intersection for 1 ≤ i ≤ n‐1, then □_i=1ⁿ G_i is a core. We then apply the results to construct cores that are box products with Mycielski construction factors or with odd graph factors. We also show that K(r,2r+1) □ C_2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r,2r+1) □ C_2l+1 is a core for r > l ≥ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

NP completeness of the edge precoloring extension problem on bipartite graphs

We show that the following problem is NP complete: Let G be a cubic bipartite graph and f be a precoloring of a subset of edges of G using at most three colors. Can f be extended to a proper edge 3‐coloring of the entire graph G? This result provides a natural counterpart to classical Holyer's result on edge 3‐colorability of cubic graphs and a strengthening of results on precoloring extension of perfect graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 156–160, 2003     

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     

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.     

Spanning 2‐trails from degree sum conditions

Suppose G is a simple connected n‐vertex graph. Let σ₃(G) denote the minimum degree sum of three independent vertices in G (which is ∞ if G has no set of three independent vertices). A 2‐trail is a trail that uses every vertex at most twice. Spanning 2‐trails generalize hamilton paths and cycles. We prove three main results. First, if σ₃G)≥ n ‐ 1, then G has a spanning 2‐trail, unless G ? K_1,3. Second, if σ₃(G) ≥ n, then G has either a hamilton path or a closed spanning 2‐trail. Third, if G is 2‐edge‐connected and σ₃(G) ≥ n, then G has a closed spanning 2‐trail, unless G ? K_2,3 or K (the 6‐vertex graph obtained from K_2,3 by subdividing one edge). All three results are sharp. These results are related to the study of connected and 2‐edge‐connected factors, spanning k‐walks, even factors, and supereulerian graphs. In particular, a closed spanning 2‐trail may be regarded as a connected (and 2‐edge‐connected) even [2,4]‐factor. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 298–319, 2004     

Equitable edge‐colorings of simple graphs

An edge‐coloring of a graph G is equitable if, for each v∈V(G), the number of edges colored with any one color incident with v differs from the number of edges colored with any other color incident with v by at most one. A new sufficient condition for equitable edge‐colorings of simple graphs is obtained. This result covers the previous results, which are due to Hilton and de Werra, verifies a conjecture made by Hilton recently, and substantially extends it to a more general class of graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:175‐197, 2011     

Circular flows of nearly Eulerian graphs and vertex‐splitting

The odd edge connectivity of a graph G, denoted by λ_o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and ? be a positive real number. We proved that there is a function f_S(?) (depends on the surface S and lim_?→0 f_S(?)=∞) such that any graph G embedded in S with the odd‐edge connectivity at least f_S(?) admits a nowhere‐zero circular (2+?)‐flow. Another major result of the work is a new vertex splitting lemma which maintains the old edge connectivity and graph embedding. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 147–161, 2002     

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     

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     

Polychromatic colorings of bounded degree plane graphs

A polychromatic k‐coloring of a plane graph G is an assignment of k colors to the vertices of G such that every face of G has all k colors on its boundary. For a given plane graph G, one seeks the maximum number k such that G admits a polychromatic k ‐coloring. In this paper, it is proven that every connected plane graph of order at least three, and maximum degree three, other than K₄ or a subdivision of K₄ on five vertices, admits a 3‐coloring in the regular sense (i.e., no monochromatic edges) that is also a polychromatic 3‐coloring. Our proof is constructive and implies a polynomial‐time algorithm. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 269‐283, 2009     

On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

Large Kr‐free subgraphs in Ks‐free graphs and some other Ramsey‐type problems

In this paper we present three Ramsey‐type results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph K_s on s vertices. More than 40 years ago Erd?s and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete graph K_r. Our first result provides a new lower bound for this problem, which improves previous results of various researchers. It also allows us to solve some special cases of a closely related question posed by Erd?s. For two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any red‐blue coloring of the edges of the complete graph K_N, contains either a red copy of G or a blue copy of H. The book with n pages is the graph B_n consisting of n triangles sharing one edge. Here we study the book‐complete graph Ramsey numbers and show that R(B_n, K_n) ≤ O(n³/log^3/2n), improving the bound of Li and Rousseau. Finally, motivated by a question of Erd?s, Hajnal, Simonovits, Sós, and Szemerédi, we obtain for all 0 < δ < 2/3 an estimate on the number of edges in a K₄‐free graph of order n which has no independent set of size n^1‐δ. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

On total 9‐coloring planar graphs of maximum degree seven

Given a graph G, a total k‐coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. If Δ(G) is the maximum degree of G, then no graph has a total Δ‐coloring, but Vizing conjectured that every graph has a total (Δ + 2)‐coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing that every planar (and projective) graph with Δ ≤ 7 has a total 9‐coloring by means of the discharging method. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 67–73, 1999