Edge colorings of graphs embeddable in a surface of low genus

In this paper, by using the Discharging Method, we show that any graph with maximum degree Δ 8 that is embeddable in a surface Σ of characteristic χ(Σ) 0 is class one and any graph with maximum degree Δ 9 that is embeddable in a surface Σ of characteristic χ(Σ) = − 1 is class one. For surfaces of characteristic 0 or −1, these results improve earlier results of Mel'nikov.     

Finding the exact bound of the maximum degrees of class two graphs embeddable in a surface of characteristic

In this paper, we consider the problem of determining the maximum of the set of maximum degrees of class two graphs that can be embedded in a surface. For each surface Σ, we define Δ(Σ)=max{Δ(G)| G is a class two graph of maximum degree Δ that can be embedded in Σ}. Hence Vizing's Planar Graph Conjecture can be restated as Δ(Σ)=5 if Σ is a plane. We show that Δ(Σ)=7 if (Σ)=−1 and Δ(Σ)=8 if (Σ){−2,−3}.     

Coloring edges of embedded graphs

In this paper, we prove that any graph G with maximum degree , which is embeddable in a surface Σ of characteristic χ(Σ) ≤ 1 and satisfies , is class one. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 197–205, 2000     

A note on transformations of edge colorings of bipartite graphs

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Δ and let t,n be integers, tnΔ. Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t>Δ then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzig's problem for bipartite graphs, J. Combin. Theory Ser. B 74 (1998) 160–168] for transformation of f to g in the case when t=n=Δ and G is regular.     

Oriented list colorings of graphs

A 2‐assignment on a graph G = (V,E) is a collection of pairs L(v) of allowed colors specified for all vertices v ∈V. The graph G (with at least one edge) is said to have oriented choice number 2 if it admits an orientation which satisfies the following property: For every 2‐assignment there exists a choice c(v)∈L(v) for all v ∈V such that (i) if c(v) = c(w), then vw ∉ E, and (ii) for every ordered pair (a,b) of colors, if some edge oriented from color a to color b occurs, then no edge is oriented from color b to color a. In this paper we characterize the following subclasses of graphs of oriented choice number 2: matchings; connected graphs; graphs containing at least one cycle. In particular, the first result (which implies that the matching with 11 edges has oriented choice number 2) proves a conjecture of Sali and Simonyi. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 217–229, 2001     

A note on Vizing's independence number conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then α(G)?n/2, where α(G) is the independence number of G. In this note, we verify this conjecture for n?2Δ.     

Gallai colorings of non-complete graphs

Gallai-colorings of complete graphs-edge colorings such that no triangle is colored with three distinct colors-occur in various contexts such as the theory of partially ordered sets (in Gallai’s original paper), information theory and the theory of perfect graphs. We extend here Gallai-colorings to non-complete graphs and study the analogue of a basic result-any Gallai-colored complete graph has a monochromatic spanning tree-in this more general setting. We show that edge colorings of a graph H without multicolored triangles contain monochromatic connected subgraphs with at least (α(H)²+α(H)−1)⁻¹|V(H)| vertices, where α(H) is the independence number of H. In general, we show that if the edges of an r-uniform hypergraph H are colored so that there is no multicolored copy of a fixed F then there is a monochromatic connected subhypergraph H₁⊆H such that |V(H₁)|≥c|V(H)| where c depends only on F, r, and α(H).     

Finding Δ(Σ) for a surface σ of characteristic χ(Σ) = −5

For each surface Σ, we define Δ(Σ) = max{Δ(G)|Gis a class two graph of maximum degree Δ(G) that can be embedded in Σ}. Hence, Vizing's Planar Graph Conjecture can be restated as Δ(Σ) = 5 if Σ is a plane. In this paper, we show that Δ(Σ) = 9 if Σ is a surface of characteristic χ(Σ) = ?5. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:148‐168, 2011     

Some results in square-free and strong square-free edge-colorings of graphs

The set of problems we consider here are generalizations of square-free sequences [A. Thue, Über unendliche Zeichenreichen, Norske Vid Selsk. Skr. I. Mat. Nat. Kl. Christiana 7 (1906) 1-22]. A finite sequence a₁a₂…a_n of symbols from a set S is called square-free if it does not contain a sequence of the form ww=x₁x₂…x_mx₁x₂…x_m,x_i∈S, as a subsequence of consecutive terms. Extending the above concept to graphs, a coloring of the edge set E in a graph G(V,E) is called square-free if the sequence of colors on any path in G is square-free. This was introduced by Alon et al. [N. Alon, J. Grytczuk, M. Ha?uszczak, O. Riordan, Nonrepetitive colorings of graphs, Random Struct. Algor. 21 (2002) 336-346] who proved bounds on the minimum number of colors needed for a square-free edge-coloring of G on the class of graphs with bounded maximum degree and trees. We discuss several variations of this problem and give a few new bounds.     

Finding for a Surface Σ of Characteristic −4

For each surface Σ, we define max G is a class two graph of maximum degree that can be embedded in . Hence, Vizing's Planar Graph Conjecture can be restated as if Σ is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that if Σ is a surface of characteristic . Until now, all known satisfy . This is the first case where .     

Vertex colorings of graphs without short odd cycles

Motivated by the work of Ne?et?il and Rödl on “Partitions of vertices” we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r‐coloring of the vertices of H yields a monochromatic and induced copy of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 255‐264, 2011     

On splittable colorings of graphs and hypergraphs

The notion of a split coloring of a complete graph was introduced by Erd?s and Gyárfás [ 7 ] as a generalization of split graphs. In this work, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two‐round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)‐split coloring problem of [ 7 ] are closer in nature to the Turán theory of graphs rather than Ramsey theory. We extend the notion of these colorings to hypergraphs and provide bounds and some exact results. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 226–237, 2002     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

Rainbow structures in locally bounded colorings of graphs

We study approximate decompositions of edge‐colored quasirandom graphs into rainbow spanning structures: an edge‐coloring of a graph is locally ‐bounded if every vertex is incident to at most edges of each color, and is (globally) ‐bounded if every color appears at most times. Our results imply the existence of: (1) approximate decompositions of properly edge‐colored into rainbow almost‐spanning cycles; (2) approximate decompositions of edge‐colored into rainbow Hamilton cycles, provided that the coloring is ‐bounded and locally ‐bounded; and (3) an approximate decomposition into full transversals of any array, provided each symbol appears times in total and only times in each row or column. Apart from the logarithmic factors, these bounds are essentially best possible. We also prove analogues for rainbow ‐factors, where is any fixed graph. Both (1) and (2) imply approximate versions of the Brualdi‐Hollingsworth conjecture on decompositions into rainbow spanning trees.     

Asymmetric 2‐colorings of graphs

We show that the edges of every 3‐connected planar graph except K₄ can be colored with two colors in such a way that the graph has no color‐preserving automorphisms. Also, we characterize all graphs that have the property that their edges can be 2‐colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface that induces a nontrivial color‐preserving automorphism of the graph.     

On the connectivity of proper colorings of random graphs and hypergraphs

Let Ω_q=Ω_q(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γ_q be the graph with vertex set Ω_q where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_n,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γ_q is connected if d is sufficiently large and . This is optimal up to the first order in d. Furthermore, with a few more colors, we find that the diameter of Γ_q is O(n) w.h.p., where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber dynamics Markov Chain on Ω_q is ergodic w.h.p.     

The size of edge chromatic critical graphs with maximum degree 6

In 1968, Vizing [Uaspekhi Mat Nauk 23 (1968) 117–134; Russian Math Surveys 23 (1968), 125–142] conjectured that for any edge chromatic critical graph with maximum degree , . This conjecture has been verified for . In this article, by applying the discharging method, we prove the conjecture for . © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 149–171, 2009     

Orientable edge colorings of graphs

An edge coloring of a graph is orientable if and only if it is possible to orient the edges of the graph so that the color of each edge is determined by the head of its corresponding oriented arc. The goals of this paper include finding a forbidden substructure characterization of orientable colorings and giving a linear time recognition algorithm for orientable colorings.An edge coloring is lexical if and only if it is possible to number the vertices of the graph so that the color of each edge is determined by its lower endpoint. Lexical colorings are, of course, the orientable colorings in which the underlying orientation is acyclic. Lexical colorings play an important role in Canonical Ramsey theory, and it is this standpoint that motivates the current study.     

Distinguishing colorings of Cartesian products of complete graphs

We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph K_s,t which admits only the identity automorphism. In particular, this allows us to determine the distinguishing number of the Cartesian product of complete graphs.