The odd girth of a graph \(G\) gives the length of a shortest odd cycle in \(G\). Let \(f(k, g)\) denote the smallest \(n\) such that there exists a \(k\)-regular graph of order \(n\) and odd girth \(g\). It is known that \(f(k, g) \geqslant k g / 2\) and that \(f(k, g)=k g / 2\) if \(k\) is even. T
Extremal graphs with prescribed covering number
β Scribed by J.A Bondy
- Publisher
- Elsevier Science
- Year
- 1978
- Tongue
- English
- Weight
- 120 KB
- Volume
- 24
- Category
- Article
- ISSN
- 0095-8956
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## Abstract We determine the minimum number of edges in a regular connected graph on __n__ vertices, containing a complete subgraph of order __k__ β€ __n__/2. This enables us to confirm and strengthen a conjecture of P. ErdΓΆs on the existence of regular graphs with prescribed chromatic number.
Let G be a simple graph of order n(G). A vertex set D of G is dominating if every vertex not in D is adjacent to some vertex in D, and D is a covering if every edge of G has at least one end in D. The domination number 7(G) is the minimum order of a dominating set, and the covering number/~(G) is th