Affiliation:
1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
2. Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
Abstract
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.
Funder
National Science Foundation
National Natural Science Foundation of China
Quantum Information Technologies
Publisher
Oxford University Press (OUP)
Reference26 articles.
1. Cycles modulo k;Bollobás;Bull. London Math. Soc.,1977
2. Cycles in a graph whose lengths differ by one or two;Bondy;J. Graph Theory,1998
3. Graphs with a cycle of length divisible by three;Chen;J. Combin. Theory Ser. B,1994
4. Minimum degree conditions for the existence of cycles of all lengths modulo $k$ in graphs;Chiba
5. Which graphs are pancyclic modulo k;Dean,1988
Cited by
6 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献