Strengthening theorems of Dirac and Erd\H{o}s on disjoint cycles

TL;DR

This paper extends classical theorems by establishing new conditions under which graphs contain k disjoint cycles, considering degree constraints and the number of disjoint triangles, thus broadening the understanding of cycle packings.

Contribution

It generalizes the Dirac-Erdős theorem by incorporating bounds on disjoint triangles and degree differences, providing new sufficient conditions for k disjoint cycles.

Findings

Graphs with degree difference ≥ 3k contain k disjoint cycles.

In graphs with at most t disjoint triangles, degree difference ≥ 2k + t guarantees k disjoint cycles.

The results unify and extend previous theorems like the Corrádi-Hajnal theorem.

Abstract

Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erd\H{o}s proved that $G$ contains $k$ (vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \ge k^{2} + 2k - 4$. The main result of this paper is that for $k \ge 2$, every graph $G$ with $|V(G)| \ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)| - |L_{k}(G)| \ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k \ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \ge 3k$, then $G$ contains $k$ disjoint cycles. This generalizes the Corr\'{a}di-Hajnal Theorem, which states that every graph $G$ with $H_{k}(G) = V(G)$ and $|H_{k}(G)| \ge 3k$ contains $k$ disjoint cycles.