On the parameter $\mu_{21}$ of a complete bipartite graph

TL;DR

This paper determines the exact value of the parameter _{21} for complete bipartite graphs, which measures the number of vertices with interval spectra in a game-theoretic edge coloring scenario.

Contribution

It provides the first exact calculation of _{21} for all complete bipartite graphs, advancing understanding of interval spectra in asymmetric coloring games.

Findings

Exact _{21} values for all K_{m,n}

Characterization of optimal strategies for Alice and Bob

Insights into interval spectra in bipartite graphs

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. Suppose that all edges of a graph $G$ are colored in the game of Alice and Bob with asymmetric distribution of roles. Alice determines the number $t$ of colors in the future proper edge coloring of $G$ and aspires to minimize the number of vertices with an interval spectrum in it. Bob colors edges of $G$ with $t$ colors and aspires to maximize that number. $\mu_{21}(G)$ is equal to the number of vertices of $G$ with an interval spectrum at the finish of the game on the supposition that both players choose their best strategies. In this paper, for arbitrary positive integers $m$ and $n$, the exact value of the parameter $\mu_{21}(K_{m,n})$ is found.