The minimal trivalent graphs with given smallest odd cycle
✍ Scribed by Peter Kovács
- Publisher
- Elsevier Science
- Year
- 1985
- Tongue
- English
- Weight
- 313 KB
- Volume
- 54
- Category
- Article
- ISSN
- 0012-365X
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📜 SIMILAR VOLUMES
This note presents a solution to the following problem posed by Chen, Schelp, and Soltés: find a simple graph with the least number of vertices for which only the degrees of the vertices that appear an odd number of times are given.
## Abstract The odd girth of a graph __G__ is the length of a shortest odd cycle in __G__. Let __d__(__n, g__) denote the largest __k__ such that there exists a __k__‐regular graph of order __n__ and odd girth __g__. It is shown that __d____n, g__ ≥ 2|__n__/__g__≥ if __n__ ≥ 2__g__. As a consequenc
We show that every graph G on n vertices with minimal degree at least n / k contains a cycle of length at least [ n / ( k -111. This verifies a conjecture of Katchalski. When k = 2 our result reduces t o the classical theorem of Dirac that asserts that if all degrees are at least i n then G is Hamil