Disjoint cycles of different lengths in graphs and digraphs

TL;DR

This paper investigates conditions under which graphs and digraphs contain multiple vertex-disjoint cycles of distinct lengths, providing tight bounds for undirected graphs and verifying a conjecture for specific classes of digraphs.

Contribution

It establishes optimal degree bounds for undirected graphs to contain disjoint cycles of different lengths and verifies a conjecture for certain classes of directed graphs.

Findings

Minimum degree bounds for undirected graphs with disjoint cycles of different lengths.

Verification of Lichiardopol's conjecture for tournaments and specific digraph classes.

Degree conditions ensuring the existence of cycles with prescribed length properties.

Abstract

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider stronger situations, and exhibit degree bounds (some of which are best possible) when e.g. the graph is triangle-free, or the $k$ cycles are requested to have different lengths congruent to some values modulo some $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for regular digraphs and digraphs of small order.