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Bounding the list color function threshold from above

Hemanshu Kaul, Akash Kumar, Andrew Liu, Jeffrey A. Mudrock, Patrick Rewers, Paul Shin, Michael Scott Tanahara and Khue To

Vol. 16 (2023), No. 5, 849–882
Abstract

The chromatic polynomial of a graph G, denoted by P(G,m), is equal to the number of proper m-colorings of G for each m . In 1990, Kostochka and Sidorenko introduced the list color function of graph G, denoted by P(G,m), which is a list analogue of the chromatic polynomial. The list color function threshold of G, denoted by τ(G), is the smallest k such that P(G,k) > 0 and P(G,m) = P(G,m) whenever m k. It is known that for every graph G, τ(G) is finite, and a recent paper of Kaul et al. suggests that complete bipartite graphs may be the key to understanding the extremal behavior of τ. We develop tools for bounding the list color function threshold of complete bipartite graphs from above. We show that, for any n 2, τ(K2,n) (n + 2.05)1.24. Interestingly, our proof makes use of classical results such as Rolle’s theorem and Descartes’ rule of signs.

Keywords
list coloring, chromatic polynomial, list color function, list color function threshold, enumerative chromatic-choosability
Mathematical Subject Classification
Primary: 05C15, 05C30
Supplementary material

Python code

Milestones
Received: 17 July 2022
Revised: 5 November 2022
Accepted: 9 November 2022
Published: 9 December 2023

Communicated by Ronald Gould
Authors
Hemanshu Kaul
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL
United States
Akash Kumar
Department of Mathematics
College of Lake County
Grayslake, IL
United States
Andrew Liu
Department of Mathematics
College of Lake County
Grayslake, IL
United States
Jeffrey A. Mudrock
Department of Mathematics
College of Lake County
Grayslake, IL
United States
Patrick Rewers
Department of Mathematics
College of Lake County
Grayslake, IL
United States
Paul Shin
Department of Mathematics
College of Lake County
Grayslake, IL
United States
Michael Scott Tanahara
Department of Mathematics
College of Lake County
Grayslake, IL
United States
Khue To
Department of Mathematics
College of Lake County
Grayslake, IL
United States