## Abstract The topic of this paper is representing permutation groups by connected graphs with proper edge colourings. Every connected graph __G__ with a proper edge colouring ฯ determines a group __A~c~__(__G__, ฯ) of graph automorphisms which preserve the colours of the edges. We characterize pe
Representing Groups by Colourings of Graphs
โ Scribed by Ulrike Baumann
- Publisher
- John Wiley and Sons
- Year
- 1997
- Tongue
- English
- Weight
- 297 KB
- Volume
- 188
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
## Abstract A perfect colouring ฮฆ of a simple undirected connected graph __G__ is an edge colouring such that each vertex is incident with exactly one edge of each colour. This paper concerns the problem of representing groups by graphs with perfect colourings. We define groups of graph automorphis
It is proved that if G is a planar graph with total (vertex-edge) chromatic number ฯ , maximum degree and girth g, then ฯ = + 1 if โฅ 5 and g โฅ 5, or โฅ 4 and g โฅ 6, or โฅ 3 and g โฅ 10. These results hold also for graphs in the projective plane, torus and Klein bottle.
This paper proves that if a graph G has an orientation D such that for each cycle C with djCj รฐmod kร 2 f1; 2; . . . ; 2d ร 1g we have jCj=jC รพ j4k=d and jCj=jC ร j4k=d; then G has a รฐk; dร-colouring and hence w c รฐGร4k=d: This is a generalization of a result of Tuza (J. Combin. Theory Ser. B 55 (19
For a large class of finite Cayley graphs we construct covering graphs whose automorphism groups coincide with the groups of lifted automorphisms. As an application we present new examples of 1ร2-transitive and 1-regular graphs.
Let G be a permutation group of finite degree d. We prove that the product of the orders of the composition factors of G that are not alternating groups acting naturally, in a sense that will be made precise, is bounded by c d-1 /d, where c = 4 5. We use this to prove that any quotient G/N of G has