Long heterochromatic paths in heterochromatic triangle free graphs

Details Long heterochromatic paths in heterochromatic triangle free graphs Long heterochromatic paths in heterochromatic triangle free graphs

2008-04-29

arxiv.org/abs/0804.4526v1

Mathematics

Combinatorics

12 pages

Scientific paper

In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum color degree at least $k$. For the heterochromatic triangle free graphs $K_n$, we obtain that for every vertex $v\in V(K_n)$, $K_n$ has a heterochromatic $v$-path of length at least $d^c(v)$; whereas for the heterochromatic triangle free graphs $G$ we show that if, for any vertex $v\in V(G)$, $d^c(v)\geq k\geq 6$, then $G$ a heterochromatic path of length at least $\frac{3k}{4}$.

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