Multigraphs with quasiweak odd cycles

In this paper we define a class of multigraphs with chromatic index equal to the maximum degree d. They are characterized by a property of their elementary odd cycles. It is shown that these graphs are panchromatic (i.e., they have a good k-coloring for any k). In the partially ordered set of color-feasible sequences of these graphs, all maximal sequences have at most d + 1 terms.     

On the existence of generalized good and equitable edge colorings

Some classes of graphs are described which are extensions of bipartite multigraphs. Exclusion of some specific partial subgraphs gives some properties of edge colorability. in particular sufficient conditions are developed for the existence of generalized good and equitable colorings.     

Regular and canonical colorings

A general framework for coloring problems is described; the concept of regular coloring is introduced; it simply means that one specifies in each edge the maximum and the minimum number of nodes which may have the same color.For several types of regular colorings, one defines canonical colorings where colors form an ordered set and where one always tries to use first the “smallest” colors. It is shown that for some classes of multigraphs including bipartite multigraphs, regular edge colorings corresponding to maximal color feasible sequences are canonical.     

Path colorings in bipartite graphs

The theorem of König on edge colorings in bipartite multigraphs can be seen as the integral version of the theorem of Birkhoff and von Neumann on bistochastic matrices.Here we consider the more general case where the matrix A=(a_ij) to be decomposed has real entries (instead of non negative entries). We shall concentrate on the integral case. Interpretation in terms of arc and path colorings are given with some properties of these decompositions and one shows that some balancing problems which are trivial in the classical case are now NP-complete. We also introduce requirements on the parity of the paths in the decompositions.     

On the equality of the partial Grundy and upper ochromatic numbers of graphs

A (proper) k-coloring of a graph G is a partition Π={V₁,V₂,…,V_k} of V(G) into k independent sets, called color classes. In a k-coloring Π, a vertex v∈V_i is called a Grundy vertex if v is adjacent to at least one vertex in color class V_j, for every j, j<i. A k-coloring is called a Grundy coloring if every vertex is a Grundy vertex. A k-coloring is called a partial Grundy coloring if every color class contains at least one Grundy vertex. In this paper we introduce partial Grundy colorings, and relate them to parsimonious proper colorings introduced by Simmons in 1982.     

On list incidentor (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">l</Emphasis>)-coloring

A proper incidentor coloring is called a (k, l)-coloring if the difference between the colors of the final and initial incidentors ranges between k and l. In the list variant, the extra restriction is added: the color of each incidentor must belong to the set of admissible colors of the arc. In order to make this restriction reasonable we assume that the set of admissible colors for each arc is an integer interval. The minimum length of the interval that guarantees the existence of a list incidentor (k, l)-coloring is called a list incidentor (k, l)-chromatic number. Some bounds for the list incidentor (k, l)-chromatic number are proved for multigraphs of degree 2 and 4.     

A short proof for a generalization of Vizing's theorem

For a simple graph of maximum degree Δ, it is always possible to color the edges with Δ + 1 colors (Vizing); furthermore, if the set of vertices of maximum degree is independent, Δ colors suffice (Fournier). In this article, we give a short constructive proof of an extension of these results to multigraphs. Instead of considering several color interchanges along alternating chains (Vizing, Gupta), using counting arguments (Ehrenfeucht, Faber, Kierstead), or improving nonvalid colorings with Fournier's Lemma, the method of proof consists of using one single easy transformation, called “sequential recoloring”, to augment a partial k-coloring of the edges.     

The deficiency of a regular graph

A proper edge coloring c:E(G)→Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1?k?4 and bounds for larger k.     

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

Smallest 3-graphs having a 3-colored edge in every k-coloring

How many edges must a 3-graph have if, for every k-coloring, there exist a 3-colored edge? Exact and approximate values are given for this function of k and of the number of vertices.     

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.     

An adaptive memory algorithm for the k-coloring problem

Let G=(V,E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,…,k}) to each vertex of G so that no edge has both endpoints with the same color. The adaptive memory algorithm is a hybrid evolutionary heuristic that uses a central memory. At each iteration, the information contained in the central memory is used for producing an offspring solution which is then possibly improved using a local search algorithm. The so obtained solution is finally used to update the central memory. We describe in this paper an adaptive memory algorithm for the k-coloring problem. Computational experiments give evidence that this new algorithm is competitive with, and simpler and more flexible than, the best known graph coloring algorithms.     

Breaking the rhythm on graphs

We study graph colorings avoiding periodic sequences with large number of blocks on paths. The main problem is to decide, for a given class of graphs F, if there are absolute constants t,k such that any graph from the class has a t-coloring with no k identical blocks in a row appearing on a path. The minimum t for which there is some k with this property is called the rhythm threshold of F, denoted by t(F). For instance, we show that the rhythm threshold of graphs of maximum degree at most d is between (d+1)/2 and d+1. We give several general conditions for finiteness of t(F), as well as some connections to existing chromatic parameters. The question whether the rhythm threshold is finite for planar graphs remains open.     

On DP-coloring of graphs and multigraphs

While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K _k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ _DP (G) resemble the properties of the list chromatic number χ _l (G) but some differ quite a lot. It is always the case that χ _DP (G) ≥ χ _l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.     

On 2-factor hamiltonian regular bipartite graphs

A k-regular bipartite graph is said to be 2-factor hamiltonian if each of its 2-factor is hamiltonian. It is well known that if a k-regular bipartite graph is 2-factor hamiltonian, then k?Q3. In this paper, we give a new proof of this fact.     

Sharpening an Ore-type version of the Corrádi–Hajnal theorem

Corrádi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all \(k\ge 1\) and \(n\ge 3k\), every (simple) graph G on n vertices with minimum degree \(\delta (G)\ge 2k\) contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corrádi–Hajnal theorem: For all \(k\ge 1\) and \(n\ge 3k\), every graph G on n vertices contains k disjoint cycles, provided that \(d(x)+d(y)\ge 4k-1\) for all distinct nonadjacent vertices x, y. Very recently, it was refined for \(k\ge 3\) and \(n\ge 3k+1\): If G is a graph on n vertices such that \(d(x)+d(y)\ge 4k-3\) for all distinct nonadjacent vertices x, y, then G has k vertex-disjoint cycles if and only if the independence number \(\alpha (G)\le n-2k\) and G is not one of two small exceptions in the case \(k=3\). But the most difficult case, \(n=3k\), was not handled. In this case, there are more exceptional graphs, the statement is more sophisticated, and some of the proofs do not work. In this paper we resolve this difficult case and obtain the full picture of extremal graphs for the Ore-type version of the Corrádi–Hajnal theorem. Since any k disjoint cycles in a 3k-vertex graph G must be 3-cycles, the existence of such k cycles is equivalent to the existence of an equitable k-coloring of the complement of G. Our proof uses the language of equitable colorings, and our result can be also considered as an Ore-type version of a partial case of the Chen–Lih–Wu Conjecture on equitable colorings.     

Feasible Graphs and Colorings

The problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an effective degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring.     

Disconnected Colors in Generalized Gallai‐Colorings

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. A basic property of Gallai‐colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.     

Total Edge Irregularity Strength of Toroidal Fullerene

A toroidal fullerene (toroidal polyhex) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. An edge irregular total k-labeling of a graph G is such a labeling of the vertices and edges with labels 1, 2, … , k that the weights of any two different edges are distinct, where the weight of an edge is the sum of the label of the edge itself and the labels of its two endvertices. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength, tes(G). In this paper we determine the exact value of the total edge irregularity strength of toroidal polyhexes.     

Neighbor sum distinguishing edge coloring of subcubic graphs

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let χ_Σ'(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges(we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) 5/2,then χ_Σ'(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.