## Abstract We derive bounds for __f(v)__, the maximum number of edges in a graph on __v__ vertices that contains neither threeβcycles nor fourβcycles. Also, we give the exact value of __f(v)__ for all __v__ up to 24 and constructive lower bounds for all __v__ up to 200. Β© 1993 John Wiley & Sons, I
Graphs without four-cycles
β Scribed by C. R. J. Clapham; A. Flockhart; J. Sheehan
- Publisher
- John Wiley and Sons
- Year
- 1989
- Tongue
- English
- Weight
- 722 KB
- Volume
- 13
- Category
- Article
- ISSN
- 0364-9024
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β¦ Synopsis
We investigate the values of t(n), the maximum number of edges in a graph with n vertices and not containing a four-cycle. Techniques for finding these are developed and the values of t(n) for all n up to 21 are obtained. All the corresponding extremal graphs are found.
π SIMILAR VOLUMES
We prove that every bipartite C 2' -free graph G contains a C 4free subgraph H with e(H) ! e(G)=(' Γ 1). The factor 1=(' Γ 1) is best possible. This implies that ex(n; C 2' ) 2(' Γ 1)ex(n; fC 4 ; C 2' g), which settles a special case of a conjecture of Erdo Λs and Simonovits.
## Abstract Let __G__ be a toroidal graph without cycles of a fixed length __k__, and Ο~__l__~(__G__) the list chromatic number of __G__. We establish tight upper bounds of Ο~__l__~(__G__) for the following values of __k__: Β© 2009 Wiley Periodicals, Inc. J Graph Theory 65: 1β15, 2010.
It is shown that there is a constant \(c\) such that if \(G\) is a graph embedded in a surface of genus \(g\) (either orientable or non-orientable) and the length of a shortest non-bounding cycle of \(G\) is at least \(c \log (g+1)\), then \(G\) is six-colorable. A similar result holds for three- an
It is easy to see that planar graphs without 3-cycles are 3-degenerate. Recently, it was proved that planar graphs without 5-cycles are also 3-degenerate. In this paper it is shown, more surprisingly, that the same holds for planar graphs without 6-cycles.