Sharp minimum degree conditions for the existence of disjoint theta graphs

Emily Marshall¹ ; Michael Santana²

¹Arcadia University
²Grand Valley State University

The electronic journal of combinatorics, Tome 28 (2021) no. 3

Résumé

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs. Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs. While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.

Détail

DOI : 10.37236/9670

Classification : 05C70, 05C35, 05C07
Mots-clés : vertex-disjoint cycles, even cycle, theta graph, minimum degree