It is shown that the chromatic number of any graph with maximum degree d in which the number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed d 2 รf is at most O(dรlog f ). This is tight (up to a constant factor) for all admissible values of d and f.
Dense Graphs with Cycle Neighborhoods
โ Scribed by A. Seress; T. Szabo
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 420 KB
- Volume
- 63
- Category
- Article
- ISSN
- 0095-8956
No coin nor oath required. For personal study only.
โฆ Synopsis
For all (\varepsilon>0), we construct graphs with (n) vertices and (>n^{2-n}) edges, for arbitrarily large (n), such that the neighborhood of each vertex is a cycle. This result is asymptotically best possible. "1995 Academic Press. Inc.
๐ SIMILAR VOLUMES
## Abstract A description is obtained for connected graphs in which a point __u__ is adjacent of __v__ only if __u__ is adjacent to all points whose degree is greater than that of __v__. The minimum number of lines in such a grpah with all points having degree at least __d__ is also determined. Fin
## Abstract Let __G__ be a graph of order __n__ and define __NC(G)__ = min{|__N__(__u__) โช __N__(__v__)| |__uv__ โ __E__(__G__)}. A cycle __C__ of __G__ is called a __dominating cycle__ or __D__โ__cycle__ if __V__(__G__) โ __V__(__C__) is an independent set. A __D__โ__path__ is defined analogously.
We present and prove several results concerning the length of longest cycles in 2connected or I-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings.
## Abstract For several years, the study of neighborhood unions of graphs has given rise to important structural consequences of graphs. In particular, neighborhood conditions that give rise to hamiltonian cycles have been considered in depth. In this paper we generalize these approaches to give a