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 non-bipartite regular graphs of girth 4
β Scribed by Zhao Dong-fang
- Publisher
- Elsevier Science
- Year
- 1989
- Tongue
- English
- Weight
- 95 KB
- Volume
- 47
- Category
- Article
- ISSN
- 0095-8956
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## Abstract Let __B(G)__ be the edge set of a bipartite subgraph of a graph __G__ with the maximum number of edges. Let __b~k~__ = inf{|__B(G)__|/|__E(G)__β__G__ is a cubic graph with girth at least __k__}. We will prove that lim~k β β~ __b~k~__ β₯ 6/7.
In this paper we discuss isomorphic decompositions of regular bipartite graphs into trees and forests. We prove that: (1) there is a wide class of r-regular bipartite graphs that are decomposable into any tree of size r, (2) every r-regular bipartite graph decomposes into any double star of size r,
Let denote a bipartite distance-regular graph with diameter D β₯ 4 and valency k β₯ 3. Let ΞΈ 0 > ΞΈ 1 > β’ β’ β’ > ΞΈ D denote the eigenvalues of and let E 0 , E 1 , . . . , E D denote the associated primitive idempotents. Fix s (1 β€ s β€ D -1) and abbreviate E := E s . We say E is a tail whenever the entry