Year: 2024
Type: article
Abstract: The notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an incompatibility system over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐bounded if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.
Source: Journal of Graph Theory
DOI: 10.1002/jgt.23178
Authors: Xiaohan Cheng, Jie Hu, Donglei Yang
Institutions: University of Jinan, Nankai University, Shandong University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4402555833