A Unified Proof of Conjectures on Cycle Lengths in Graphs

Abstract

In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach: 1. Every graph $G$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; in addition, if $G$ is $2$-connected and non-bipartite, then it contains cycles of all lengths modulo $k$. 2. For all $k\geq 3$, every $k$-connected graph contains a cycle of length zero modulo $k$. 3. Every $3$-connected non-bipartite graph with minimum degree at least $k+1$ contains $k$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $k$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible.

A strengthening on odd cycles in graphs of given chromatic number

“…Confirming a conjecture of Erdős [5], Kostochka, Sudakov and Verstraëte [12] showed that if such G does not contain a triangle, then it contains at least Ω(k 2 log k) cycles of consecutive lengths. Recently, the authors and Liu [9] proved a conjecture of Sudakov and Verstraëte [20] that such G contains k − 1 cycles of consecutive lengths. We remark that Theorem 1.1 and these results of [9,16] are all tight for G being the clique K k+1 .…”

“…The proof for graphs without a triangle is motivated by [12] and utilizes extremal arguments, where we use a new lemma on A-B paths (see Lemma 3.2). On the other hand, the proof for graphs containing a triangle follows the line of [9] and relies on the structural analysis.…”

Divisible subdivisions

“…For illustrative purposes let us mention that, confirming a conjecture of Thomassen [12], Gao et al proved [7] that requiring minimum degree guarantees cycles of all even lengths modulo , including of course cycles of length divisible by ; requiring is necessary to guarantee a cycle of length divisible by , at least for odd , as shown by the complete bipartite graph for . In contrast, our result allows to argue about existence of cycles of length divisible by in much sparser graphs.…”

How to build a pillar: a proof of Thomassen's conjecture

“…There have been numerous powerful methods for embedding paths and cycles developed in the past three decades, such as Robertson and Seymour's work on graph linkage [25] (see also [2,26]), Bondy and Simonovits's use of Breadth First Search [3] (see also [23]), Krivelevich and Sudakov's use of Depth First Search [17] and the use of expanders in a long line of work by Krivelevich (see e.g. his survey [16] and more recently [7]), see also a recent method developed by Gao, Huo, Liu and Ma [9]. Despite these developments and the simple nature of pillars, the innocent looking conjecture of Thomassen has seen no progress in the past thirty years.…”