On one-sided interval edge colorings of biregular bipartite graphs

TL;DR

This paper investigates the minimum number of colors needed for proper edge colorings of bipartite graphs where all edges incident to vertices in one part form consecutive integer intervals, focusing on graphs with biregular bipartite structures.

Contribution

It provides estimates for the minimum number of colors required for interval edge colorings on one part of bipartite graphs, specifically in biregular cases, advancing understanding of such colorings.

Findings

Estimates for the parameter $w_R(G)$ in certain bipartite graphs.

Results applicable when $R$ is the set of all vertices in one bipartition.

Focus on biregular bipartite graphs with interval edge colorings.

Abstract

A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called a spectrum of $x$. An arbitrary nonempty subset of consecutive integers is called an interval. We say that a proper edge $t$-coloring of a graph $G$ is interval in the vertex $x$ if the spectrum of $x$ is an interval. We say that a proper edge $t$-coloring $\varphi$ of a graph $G$ is interval on a subset $R_0$ of vertices of $G$, if for an arbitrary $x\in R_0$, $\varphi$ is interval in $x$. We say that a subset $R$ of vertices of $G$ has an $i$-property if there is a proper edge $t$-coloring of $G$ which is interval on $R$. If $G$ is a graph, and a subset $R$ of its vertices has an $i$-property, then the minimum value of $t$ for which there is a proper edge $t$-coloring of $G$ interval on $R$ is denoted by $w_R(G)$. In this paper, for some bipartite graphs, we estimate the value of this parameter in that cases when $R$ coincides with the set of all vertices of one part of the graph.