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The coloring problem is a well-research topic and its complexity is known for several classes of graphs. However, the question of its complexity remains open for the class of antiprismatic graphs, which are the complement of prismatic…

In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A…

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An \emph{interval total $t$-coloring} of a graph $G$ is a total…

In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out…

Graph coloring is fundamental to distributed computing. We give the first sub-logarithmic distributed algorithm for coloring cluster graphs. These graphs are obtained from the underlying communication network by contracting nodes and edges,…

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We…

The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes -- a purely combinatorial one and two geometric ones. It is shown, that most…

A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a…

For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings.…

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high…

A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show…

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…

A proper edge-coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval of consecutive integers. Interval thickness s(G) of a graph G is the smallest number of interval colorable graphs…

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed…

In parallel computing, a valid graph coloring yields a lock-free processing of the colored tasks, data points, etc., without expensive synchronization mechanisms. However, coloring is not free and the overhead can be significant. In…

Given a graph $G$ and a positive integer $k$, the 2-Load coloring problem is to check whether there is a $2$-coloring $f:V(G) \rightarrow \{r,b\}$ of $G$ such that for every $i \in \{r,b\}$, there are at least $k$ edges with both end…

In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number, the Thue chromatic number, the harmonious chromatic number and the clique chromatic number of $P_4$-tidy graphs and…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

In an undirected graph, a conflict-free coloring (with respect to open neighborhoods) is an assignment of colors to the vertices of the graph $G$ such that every vertex in $G$ has a uniquely colored vertex in its open neighborhood. The…