# A Unified Proof of Conjectures on Cycle Lengths in Graphs

@article{Gao2019AUP,
title={A Unified Proof of Conjectures on Cycle Lengths in Graphs},
author={Jun-ming Gao and Qingyi Huo and Chun-Hung Liu and Jie Ma},
journal={arXiv: Combinatorics},
year={2019}
}
• Published 17 April 2019
• Mathematics
• arXiv: Combinatorics

## References

SHOWING 1-10 OF 37 REFERENCES

### Cycles in triangle-free graphs of large chromatic number

• Mathematics
Comb.
• 2017
This paper proves the stronger fact that every triangle-free graph G of chromatic number k≥k0(ε) contains cycles of 1/64(1 − ε)k2 logk/4 consecutive lengths, and a cycle of length at least 1/4(1- ε), and gives new lower bounds on the circumference and the number of different cycle lengths for k-chromatic graphs in other monotone classes.

### On Arithmetic Progressions of Cycle Lengths in Graphs

This paper proves that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths.

### Cycle lengths in sparse graphs

• Mathematics
Comb.
• 2008
The result improves all previously known lower bounds on the length of the longest cycle and shows that Ω ` d (g−1)/2 is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g.

### Extremal problems for cycles in graphs

In this survey we consider extremal problems for cycles of prescribed lengths in graphs. The general extremal problem is cast as follows: if $$\mathcal{C}$$ is a set of cycles, determine the largest

### Minimum degree conditions for the existence of cycles of all lengths modulo $k$ in graphs

• Mathematics
• 2019
Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper,

### Cycles of even lengths modulo k

• A. Diwan
• Mathematics, Computer Science
J. Graph Theory
• 2010
It is proved that for all positive integers k and m, every graph of minimum degree at least k+1 contains a cycle of length congruent to 2m modulo k if the minimum degree is at least 2k−1, which improves the previously known bound of 3k−2.

### The Extremal Function for Cycles of Length l mod k

• Mathematics
Electron. J. Comb.
• 2017
It is shown that c_k(\ell) is proportional to the largest average degree of a $C_{\ell}$-free graph on $k$ vertices, which determines $c_k(ell)$ up to an absolute constant.