On the $\mu$-parameters of the Petersen graph

TL;DR

This paper investigates specific parameters related to proper edge colorings of the Petersen graph, providing exact values for these parameters that measure the extent of interval spectra at vertices.

Contribution

The paper determines the exact values of the parameters _{11}, _{12}, _{21}, and _{22} for the Petersen graph, a problem previously unresolved.

Findings

Exact values of _{11}, _{12}, _{21}, _{22} for the Petersen graph.

Characterization of proper edge colorings with interval spectra.

Insights into the structure of edge colorings in the Petersen graph.

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 the Petersen graph, the exact values of the parameters $\mu_{11}$, $\mu_{12}$, $\mu_{21}$ and $\mu_{22}$ are found.