Some remarks on relations between the $\mu$-parameters of regular graphs

TL;DR

This paper investigates the relationships between certain parameters related to proper edge colorings of regular graphs, focusing on the spectrum of vertex incident edge colors and their extremal properties.

Contribution

It introduces and analyzes new $ ext{μ}$-parameters for regular graphs, revealing relations between these parameters and proper edge colorings.

Findings

Derived inequalities between $ ext{μ}$-parameters for regular graphs.

Established bounds for the number of vertices with interval spectra.

Identified conditions under which extremal $ ext{μ}$-parameters are achieved.

Abstract

For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi:E(G)\rightarrow \{1,...,t\}$ is called a proper edge $t$-coloring of a graph $G$, if adjacent edges are colored differently and each of $t$ colors is used. The least value of $t$ for which there exists a proper edge $t$-coloring of a graph $G$ is denoted by $\chi'(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality $\chi'(G)\leq t\leq |E(G)|$, we denote by $\alpha(G,t)$ the set of all proper edge $t$-colorings of $G$. Let us also define a set $\alpha(G)$ of all proper edge colorings of a graph $G$: $$ \alpha(G)\equiv\bigcup_{t=\chi'(G)}^{|E(G)|}\alpha(G,t). $$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $\varphi\in\alpha(G)$ and $x\in V(G)$, then the set of colors of edges of $G$ which are incident with $x$ is denoted by $S_G(x,\varphi)$ and is called a spectrum of the vertex $x$ of the graph $G$ at the proper edge coloring $\varphi$. If $G$ is a graph and $\varphi\in\alpha(G)$, then define $f_G(\varphi)\equiv|\{x\in V(G)/S_G(x,\varphi) \textrm{is an interval}\}|$. For a graph $G$ and any integer $t$, satisfying the inequality $\chi'(G)\leq t\leq |E(G)|$, we define: $$ \mu_1(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_G(\varphi),\qquad \mu_2(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_G(\varphi). $$ For any graph $G$, we set: $$ \mu_{11}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t),\qquad \mu_{12}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_1(G,t), $$ $$ \mu_{21}(G)\equiv\min_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t),\qquad \mu_{22}(G)\equiv\max_{\chi'(G)\leq t\leq|E(G)|}\mu_2(G,t). $$ For regular graphs, some relations between the $\mu$-parameters are obtained.