Hamilton cycles and paths in vertex-transitive graphs—Current directions

Hamilton cycles and paths in vertex-transitive graphs - Current directions

In this article current directions in solving Lovász’s problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given. © 2009 Elsevier B.V. All rights reserved. 1. Historical motivation In 1969, Lovász [59] asked whether every finite connected vertex-transitive graph has a Hamilton path, that is, a simple path going through all vertices, thus tying toge...

Hamilton cycles in dense vertex-transitive graphs

Vertex heaviest paths and cycles in quasi-transitive digraphs

A digraph D is called a quasi-transitive digraph (QTD) if for any triple x, y, z of distinct vertices of D such that (x, y) and (y, z) are arcs of D there is at least one arc from x to z or from z to x. Solving a conjecture by J. Bang-Jensen and J. Huang (J. Graph Theory, to appear), G. Gutin (Australas. J. Combin., to appear) described polynomial algorithms for finding a Hamiltonian cycle and ...

Vertex-Oriented Hamilton Cycles in Directed Graphs

Let D be a directed graph of order n. An anti-directed Hamilton cycle H in D is a Hamilton cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. We prove that if D is a directed graph with even order n and if the indegree and the outdegree of each vertex of D is at least 23n then D contains an anti-directed Hamilton cycle. This improves a bound of...

M-alternating Hamilton paths and M-alternating Hamilton cycles

We study M-alternating Hamilton paths, and M-alternating Hamilton cycles in a simple connected graph G on ν vertices with a perfect matchingM . Let G be a bipartite graph, we prove that if for any two vertices x and y in different parts of G, d(x)+d(y) ≥ ν/2+2, then G has an M-alternating Hamilton cycle. For general graphs, a condition for the existence of an M-alternating Hamilton path startin...