Work

Coloring Graphs With Forbidden Almost Bipartite Subgraphs

Year: 2025

Type: article

Abstract: ABSTRACT Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph , there exists a quantity such that whenever is an ‐free graph of maximum degree . The largest class of connected graphs for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph for some ). However, the optimal value for remains unknown even for such graphs. Bollobás showed, using random regular graphs, that when contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of . We improve this to a uniform constant, showing for every almost bipartite graph . This surprisingly makes the bound independent of in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP‐coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results.

Source: Random Structures and Algorithms

DOI: 10.1002/rsa.70012

Authors: James Anderson, Anton Bernshteyn, Abhishek Dhawan

Institutions: Georgia Institute of Technology, University of California, Los Angeles, University of Illinois Urbana-Champaign

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,550

Funder: National Science Foundation