On sufficient conditions for equality of the independence number and the clique cover number for a class of graphs

E. V. Prosolupov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2014), pp. 90-103

Résumé

For a simple graph sufficient condition are improved to ensure that equality of the independence number and the smallest dimension of orthonormal labeling of graph result in equality of the independence number and the clique cover number. To formulate that condition a class of graphs with certain structure is described. Let $W$ be a wheel-graph with odd number of vertices $n\geq 5$. Then delete every second edge from center vertex of the graph. This results in obtaining a structure of sequence of chordless cycles $C_4$ with a common vertex and common edges in pairs. Some properties of such a structure are examined. It is proved that every graph $H$ with property $\alpha(H) = d(H) \overline\chi(H)$ is characterized by this structure. So, if for some graph $G$ independence number is equal to smallest dimension of orthonormal labeling of $G$ and the graph $G$ is free of the described structure, then independence number of $G$ is equal to clique cover number of $G$. It discusses how the conditions have been improved in comparison with previously known conditions. Bibliogr. 17. Il. 10.

Détail

Keywords: graph, orthonormal labeling, rank, minimal rank, symmetric matrices, independent set, clique cover number, independence number, smallest dimension of orthonormal labeling.
Mots-clés : clique