Nontraceable detour graphs

https://doi.org/10.1016/j.disc.2006.07.019Get rights and content
Under an Elsevier user license
open archive

Abstract

The detour order (of a vertex v) of a graph G is the order of a longest path (beginning at v). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order. Homogeneously traceable graphs are therefore detour graphs with detour deficiency zero. In this paper, we give a number of constructions for detour graphs of all orders greater than 17 and all detour deficiencies greater than zero. These constructions are used to give examples of nontraceable detour graphs with chromatic number k, k2, and girths up to 7. Moreover we show that, for all positive integers l1 and k3, there are nontraceable detour graphs with chromatic number k and detour deficiency l.

MSC

05C38
05C12
05C75

Keywords

Longest path
Detour
Detour sequence
Girth
Bipartite graph
Homogeneously traceable

Cited by (0)

1

This material is based upon work supported by the National Research Foundation under Grant number 2053752.

2

Research supported by APVT grant agency, Grant number APVT-20-004104.