## Abstract We prove that for any planar graph __G__ with maximum degree Ξ, it holds that the chromatic number of the square of __G__ satisfies Ο(__G__^2^)ββ€β2Ξβ+β25. We generalize this result to integer labelings of planar graphs involving constraints on distances one and two in the graph. Β© 2002
The multiplicity of 1-factors in the square of a graph
β Scribed by G. R. T. Hendry
- Publisher
- John Wiley and Sons
- Year
- 1984
- Tongue
- English
- Weight
- 225 KB
- Volume
- 8
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
Several authors have shown that if G is a connected graph of even order then its square G2 has a I-factor. We show that the square of any connected graph of order 2n has at least n I-factors and describe all the extremal graphs.
π SIMILAR VOLUMES
## Abstract A 1βfactorization is constructed for the line graph of the complete graph __K~n~__ when __n__ is congruent to 0 or 1 modulo 4.
## Abstract The __square__ __G__^2^ of a graph __G__ is the graph with the same vertex set __G__ and with two vertices adjacent if their distance in __G__ is at most 2. Thomassen showed that every planar graph __G__ with maximum degree Ξ(__G__)β=β3 satisfies Ο(__G__^2^)ββ€β7. Kostochka and Woodall c
Extending a result by Hartman and Rosa (1985, Europ. J. Combinatorics 6, 45-48), we prove that for any Abelian group G of even order, except for G Z 2 n with n > 2, there exists a onefactorization of the complete graph admitting G as a sharply-vertex-transitive automorphism group.
A paopm graph G has no isolated points. I t s R m e y r u m b a r ( G ) i s the m i n i m p such that every 2-coloring of the edges of K contains a monochromatic G. The Ramhey m & t @ m y R(G) i s P the r (G) ' With j u s t one exception, namely Kq, we determine R(G) f o r proper graphs u i t h a t