Related papers: Certifying coloring algorithms for graphs without …

The Colouring problem is that of deciding, given a graph $G$ and an integer $k$, whether $G$ admits a (proper) $k$-colouring. For all graphs $H$ up to five vertices, we classify the computational complexity of Colouring for…

We show that the 4-coloring problem can be solved in polynomial time for graphs with no induced 5-cycle $C_5$ and no induced 6-vertex path $P_6$.

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical…

Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$…

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without…

For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring…

For $k\in \mathbb{N}$, a $k$-acyclic colouring of a graph $G$ is a function $f\colon V(G)\to \{0,1,\dots,k-1\}$ such that (i)~$f(u)\neq f(v)$ for every edge $uv$ of $G$, and (ii)~there is no cycle in $G$ bicoloured by $f$. For $k\in…

We provide a certifying algorithm for the problem of deciding whether a P5- free graph is 3-colorable by showing there are exactly six finite graphs that are P5-free and not 3-colorable and minimal with respect to this property.

A polynomial time algorithm that determines for a fixed integer k whether or not a P5-free graph can be k-colored is presented in this paper. If such a coloring exists, the algorithm will produce a valid k-coloring.

A colouring of a graph $G=(V,E)$ is a mapping $c\colon V\to \{1,2,\ldots\}$ such that $c(u)\neq c(v)$ for every two adjacent vertices $u$ and $v$ of $G$. The {\sc List $k$-Colouring} problem is to decide whether a graph $G=(V,E)$ with a…

This is the second paper in a series of two. The goal of the series is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex…

For graph classes $P_1,...,P_k$, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph $G$ can be partitioned into subsets $V_1,...,V_k$ so that $V_j$ induces a graph in the class $P_j$…

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

We investigate the List $H$-Coloring problem, the generalization of graph coloring that asks whether an input graph $G$ admits a homomorphism to the undirected graph $H$ (possibly with loops), such that each vertex $v \in V(G)$ is mapped to…

A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating…

A $k$-coloring of a graph is an assignment of integers between $1$ and $k$ to vertices in the graph such that the endpoints of each edge receive different numbers. We study a local variation of the coloring problem, which imposes further…

A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar…

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be…

We prove that graphs that do not contain a totally odd immersion of $K_t$ are $\mathcal{O}(t)$-colorable. In particular, we show that any graph with no totally odd immersion of $K_t$ is the union of a bipartite graph and a graph which…

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…