Further Results on the Maximum Number of Stars in Graphs with Forbidden Properties

Abstract

A graph $G$ is called $k$-edge-hamiltonian if every linear forest (i.e., a disjoint union of paths) with at most $k$ edges is contained in a Hamilton cycle of $G$. In 2018, Füredi, Kostochka and Luo determined the maximum number of $t$-stars in nonhamiltonian graphs, thereby extending an earlier result of Erdős. Recently, Berikkyzy, Hogenson, Kirsch and McDonald extended this line of research by determining the maximum number of $t$-stars in graphs that are not $k$-edge-hamiltonian, as well as in graphs failing to satisfy related properties such as traceability, hamiltonian-connectedness and $k$-hamiltonicity. For sufficiently large $t$, they also characterized the extremal graphs, while for smaller values of $t$, they proposed a conjecture. In this paper, we investigate this conjecture. We show that the conjecture fails at the critical value and further establish a threshold-type result describing the behavior of the extremal graphs when $t$ is close to this critical value.

9 pages, comments welcome!