Branch-bonds, two-factors in iterated line graphs and circuits in weighted graphs

Abstract

A nontrivial path is called a branch if it has only internal vertices of degree two and end vertices of degree not two. A set S of branches of a graph G is called a branch cut if the deletion of all edges and internal vertices of branches of S results in more components than G. A minimal branch cut is called a branch-bond. In this paper, we found some relationships amongst branch-bonds, weighted graphs and two-factors in iterated line graphs. We also obtained some bounded number of components in two-factor of iterated line graphs by using the concept of branch-bonds.

Keywords:

• two-factor
• line graph
• number of component
• branch-bond
• weighted graph

2000 AMS Subject Classifications::

• 05C35
• 05C38
• 05C40
• 05C45

Acknowledgements

The authors are indebted to the anonymous referees for their nice suggestion to make the present version more readable. This research is supported by Nature Science Funds of China.