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
Coloring graphs without short cycles and long induced paths
✍ Scribed by Golovach, Petr A.; Paulusma, Daniël; Song, Jian
- Book ID
- 121706652
- Publisher
- Elsevier Science
- Year
- 2014
- Tongue
- English
- Weight
- 512 KB
- Volume
- 167
- Category
- Article
- ISSN
- 0166-218X
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## Abstract It is well known that every planar graph __G__ is 2‐colorable in such a way that no 3‐cycle of __G__ is monochromatic. In this paper, we prove that __G__ has a 2‐coloring such that no cycle of length 3 or 4 is monochromatic. The complete graph __K__~5~ does not admit such a coloring. On
Motivated by the work of Nešetřil and R ödl on "Partitions of vertices" we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and ord