Sharp Minimum Degree Conditions for the Existence of Disjoint Theta Graphs

Emily Marshall,Michael Santana

doi:10.37236/9670

Abstract

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 $2k+1$.

Highlights

All graphs in this paper are simple, unless otherwise noted
Given a graph G, we use V (G) and E(G) to denote the sets of vertices and edges of G, respectively, and for a vertex v, we use v ∈ G to denote v ∈ V (G)
For a subgraph H of G, and for a vertex v ∈ G, the neighborhood of v in H is denoted by NH(v), and the number of neighbors of v in H will be written by dH(v)

Summary

Introduction

All graphs in this paper are simple, unless otherwise noted. “disjoint” is always taken to mean “vertex-disjoint.” We say “G contains H” to mean that the graph G contains the graph H as a subgraph. For all k ∈ Z+, if G is an n-vertex graph with n 3k and δ(G) 2k, G contains k disjoint cycles This result is sharp both in terms of the number of vertices as well as the minimum degree. This shows the minimum degree condition in Theorem 2 and Corollary 3 is best possible. The purpose of this paper is to address the gap when 4k < n < ck in which we have no minimum degree condition, let alone a sharp one, that guarantees the existence of k disjoint theta graphs. Our following main result extends the minimum degree condition in Theorem 2 and Corollary 3 for all n-vertex graphs with n 4k. We use a result of Kuhn, Osthus, and Treglown in [9] to discuss minimum degree sum versions, and we pose questions regarding potential future work

Sharpness

Notation

Structural Lemmas

Proof of Theorem 6

Minimum Degree Transition and Ore Versions

Therefore kk