Interval colorings of edges of a multigraph

TL;DR

This paper investigates interval edge colorings of bipartite multigraphs, establishing bounds, complexity results, and conditions for the existence of such colorings, with implications for graph coloring theory.

Contribution

It introduces new bounds for interval colorings, proves NP-completeness of recognizing continuous colorings, and provides sufficient conditions for their existence.

Findings

Existence of interval colorings for all intermediate values between bounds.

Recognition of continuous interval colorings is NP-complete.

Sufficient condition for continuous coloring based on degree inequalities.

Abstract

Let $G=(V_1(G),V_2(G),E(G))$ be a bipartite multigraph, and $R\subseteq V_1(G)\cup V_2(G)$. A proper coloring of edges of $G$ with the colors $1,\ldots,t$ is called interval (respectively, continuous) on $R$, if each color is used for at least one edge and the edges incident with each vertex $x\in R$ are colored by $d(x)$ consecutive colors (respectively, by the colors $1,\ldots,d(x))$, where $d(x)$ is a degree of the vertex $x$. We denote by $w_1(G)$ and $W_1(G)$, respectively, the least and the greatest values of $t$, for which there exists an interval on $V_1(G)$ coloring of the multigraph $G$ with the colors $1,\ldots,t$. In the paper the following basic results are obtained. \textbf{Theorem 2.} For an arbitrary $k$, $w_1(G)\leq k\leq W_1(G)$, there is an interval on $V_1(G)$ coloring of the multigraph $G$ with the colors $1,\ldots,k$. \textbf{Theorem 3.} The problem of recognition of the existence of a continuous on $V_1(G)$ coloring of the multigraph $G$ is $NP$-complete. \textbf{Theorem 4.} If for any edge $(x,y)\in E(G)$, where $x\in V_1(G)$, the inequality $d(x)\geq d(y)$ holds then there is a continuous on $V_1(G)$ coloring of the multigraph $G$. \textbf{Theorem 1.} If $G$ has no multiple edges and triangles, and there is an interval on $V(G)$ coloring of the graph $G$ with the colors $1,\ldots,k$, then $k\leq|V(G)|-1$.