On-Line Choice Number of Complete Multipartite Graphs: an Algorithmic Approach

TL;DR

This paper presents an algorithmic approach to determine the on-line choice number of complete multipartite graphs, providing unified strategies and bounds that improve understanding of their on-line chromatic-choosability.

Contribution

It introduces a unified strategy for on-line choice number analysis of complete multipartite graphs with any independence number m, and derives new bounds and conditions for on-line chromatic-choosability.

Findings

Graphs satisfying certain inequalities are on-line chromatic-choosable.

Bound on the on-line choice number for regular complete multipartite graphs.

New conditions linking graph size, chromatic number, and on-line choosability.

Abstract

This paper studies the on-line choice number on complete multipartite graphs with independence number $m$. We give a unified strategy for every prescribed $m$. Our main result leads to several interesting consequences comparable to known results. (1) If $ k_1-\sum_{p=2}^m(\frac{p^2}{2}-\frac{3p}{2}+1)k_p\geq 0$, where $k_p$ denotes the number of parts of cardinality $p$, then $G$ is on-line chromatic-choosable. (2) If $ |V(G)|\leq\frac{m^2-m+2}{m^2-3m+4}\chi(G)$, then $G$ is on-line chromatic-choosable. (3) The on-line choice number of regular complete multipartite graphs $K_{m\star k}$ is at most $(m+\frac{1}{2}-\sqrt{2m-2})k$ for $m\geq 3$.