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We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and…
We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of \emph{component coloring of graphs}. Given a graph, the problem is to color the vertices using minimum number of colors so that…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that…
Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex's list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting…
Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the…
We study the problem of determining whether a given graph~$G=(V,E)$ admits a matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently whether~$G-M$ is bipartite). This problem is equivalent to determine whether~$G$ admits…
For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi:E(G)\rightarrow\{1,2,\ldots,t\}$ is called a proper edge $t$-coloring of a graph…
Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for…
In this paper, we consider the problem of a star coloring. In general case the problems in NP-complete. We establish the star chromatic number for splitting graph of complete and complete bipartite graphs, as well of paths and cycles. Our…
We consider two different notions of graph colouring, namely, the $t$-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced…
Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertices of each hyperedge is partitioned into two parts, a tail…
We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree {\sl rainbow} trees in various host graphs, having their edges coloured independently and uniformly at random, using a predetermined palette.…
We define the $d$-defective incidence chromatic number of a graph, generalizing the notion of incidence chromatic number, and determine it for some classes of graphs including trees, complete bipartite graphs, complete graphs, and…
We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for…
In 1995, Galvin proved that a bipartite graph $G$ admits a list edge coloring if every edge is assigned a color list of length $\Delta(G)$, the maximum degree of the graph. This result was improved by Borodin, Kostochka and Woodall, who…
Let $H_{n,(p_m)_{m=2,\ldots,M}}$ be a random non-uniform hypergraph of dimension $M$ on $2n$ vertices, where the vertices are split into two disjoint sets of size $n$, and colored by two distinct colors. Each non-monochromatic edge of size…
Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class…
Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of switchings of two colors of vertices. It is PSPACE-complete to determine whether two given vertex $k$-colorings of a graph are…
In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most $b$ and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for…
We determine the colored patterns that appear in any $2$-edge coloring of $K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We prove the existence of a positive integer $z_2$ such that any $2$-edge coloring of…