## Abstract There are many results concerned with the hamiltonicity of __K__~1,3~‐free graphs. In the paper we show that one of the sufficient conditions for the __K__~1,3~‐free graph to be Hamiltonian can be improved using the concept of second‐type vertex neighborhood. The paper is concluded with
Hamiltonian results in K1,3-free graphs
✍ Scribed by M. M. Matthews; D. P. Sumner
- Publisher
- John Wiley and Sons
- Year
- 1984
- Tongue
- English
- Weight
- 357 KB
- Volume
- 8
- Category
- Article
- ISSN
- 0364-9024
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✦ Synopsis
There have been a number of results dealing with Hamiltonian properties in powers of graphs. In this paper we show that the square and the total graph of a K,,,-free graph are vertex pancyclic. We then discuss some of the relationships between connectivity and Hamiltonian properties in K,.3-free graphs.
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## Abstract We show that every connected __K__~1,3~‐free graph with minimum degree at least __2k__ contains a __k__‐factor and construct connected __K__~1,3~‐free graphs with minimum degree __k__ + __0__(√__k__) that have no __k__‐factor.
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We prove the following conjecture of Broersma and Veldman: A connected, locally k-connected K,,-free graph is k-hamiltonian if and only if it is (k + 2)-connected ( k L 1). We use [ 11 for basic terminology and notation, and consider simple graphs only. Let G be a graph. By V(G) and E(G) we denote,