相关论文: Vertex 2-coloring without monochromatic cycles

In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial…

This note summarizes the state of what is known about the tractability of the problem ModPath, which asks if an input undirected graph contains a simple st-path whose length satisfies modulo constraints. We also consider the problem…

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 $P_6$.

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…

A vertex coloring of a graph $G$ is called a 2-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Suppose that $G$ is a planar graph with girth $5$ and maximum degree $\Delta$. We prove…

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a…

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…

We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be…

This paper studies short brooms in edge-chromatic critical graphs. We prove that for any short broom in a $\Delta$-critical graph, at most one color is missing at more than one vertex. Moreover, this color (if exists) is missing at exactly…

Motivated by a conjecture of Liang [Y.-C. Liang. {\em Anti-magic labeling of graphs}. PhD thesis, National Sun Yat-sen University, 2013.], we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free…

A proper coloring of vertices of a graph is equitable if the sizes of any two color classes differ by at most 1. Such colorings have many applications and are interesting by themselves. In this paper, we discuss the state of art and…

Let $\vec{K}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of $\vec{K}_{\mathbb{N}}$ in which every monochromatic path has…

A vertex coloring of a graph $G$ is called a $2$-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Recently, Bousquet et al. (Discrete Mathematics, 346(4), 113288, 2023) proved that…

The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a cyclic interval $t$-coloring if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…

Given a multigraph, suppose that each vertex is given a local assignment of $k$ colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours…

The study of symmetric configurations $v_3$ with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in…

A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured…

A coloring of the vertices of a connected graph is convex if each color class induces a connected subgraph. We address the convex recoloring (CR) problem defined as follows. Given a graph $G$ and a coloring of its vertices, recolor a…

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have…