Related papers: The black-and-white coloring problem on circle gra…

In this paper we are interested in the fine-grained complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$ (the $H$-Coloring problem). The starting point is that these problems can be viewed as…

We say that an edge colouring $c$ of a graph preserves an automorphism $\varphi$ if $\varphi$ maps each edge to an edge of the same colour. Otherwise, we say that $c$ breaks $\varphi$. We call an automorphism of a graph small if it moves…

This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover…

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to…

For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the $H$-Coloring problem the graph $H$ is fixed and we ask whether an instance graph $G$ admits an $H$-coloring. A generalization of this…

In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear…

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…

Let $L$ be a set of graphs. $Free$($L$) is the set of graphs that do not contain any graph in $L$ as an induced subgraph. It is known that if $L$ is a set of four-vertex graphs, then the complexity of the coloring problem for $Free$($L$) is…

The pre-coloring extension problem consists, given a graph $G$ and a subset of nodes to which some colors are already assigned, in finding a coloring of $G$ with the minimum number of colors which respects the pre-coloring assignment. This…

For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly…

The quadrance between two points $A_1 = (x_1, y_1)$ and $A_2 = (x_2, y_2)$ is the number $Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$. Let $q$ be an odd prime power and $F_q$ be the finite field with $q$ elements. The unit-quadrance graph…

One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions.…

In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…

Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $R\cup B$ does not contain a copy of $H$? Call the largest…

A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds…

In this paper we study a problem of vertex two-coloring of undirected graph such that there is no monochromatic cycle of given length. We show that this problem is hard to solve. We give a proof by presenting a reduction from variation of…

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with…

The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic…

A {\em restraint} on a (finite undirected) graph $G = (V,E)$ is a function $r$ on $V$ such that $r(v)$ is a finite subset of ${\mathbb N}$; a proper vertex colouring $c$ of $G$ is {\em permitted} by $r$ if $c(v) \not\in r(v)$ for all…

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…