List colorings with distinct list sizes, the case of complete bipartite graphs

TL;DR

This paper investigates the sum choice number of graphs, particularly complete bipartite graphs, showing it can be bounded even as minimum degree increases, contrasting with the growth of the usual choice number.

Contribution

The paper provides tight estimates for the sum choice number of unbalanced complete bipartite graphs, revealing new bounds independent of minimum degree.

Findings

Sum choice number can be bounded while minimum degree grows.

Established tight estimates for the sum choice number of $K_{a,q}$.

Contrasts growth behaviors of sum choice number and usual choice number.

Abstract

Let $f:V \rightarrow \mathbb{N}$ be a function on the vertex set of the graph $G=(V,E)$. The graph $G$ is {\em $f$-choosable} if for every collection of lists with list sizes specified by $f$ there is a proper coloring using colors from the lists. The sum choice number, $\chi_{sc}(G)$, is the minimum of $\sum f(v)$, over all functions $f$ such that $G$ is $f$-choosable. It is known (Alon 1993, 2000) that if $G$ has average degree $d$, then the usual choice number $\chi_\ell(G)$ is at least $\Omega(\log d)$, so they grow simultaneously. In this paper we show that $\chi_{sc}(G)/|V(G)|$ can be bounded while the minimum degree $\delta_{\min}(G)\rightarrow \infty$. Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph $K_{a,q}$.