Acta Univ. Sapientiae, Informatica, 6, 1 (2014) 132{158 On vertex independence number of uniform hypergraphs Tariq A. CHISHTI Guofei ZHOU University of Kashmir Nanjing University Department of Mathematics Department of Mathematics Srinagar, India Nanjing, China email:
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[email protected] Shariefuddin PIRZADA Antal IVANYI´ University of Kashmir E¨otv¨osLor´andUniversity Department of Mathematics Faculty of Informatics Srinagar, India Budapest, Hungary email: email:
[email protected] [email protected] Abstract. Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree. 1 Introduction to independence in graphs Let n be a positive integer. A graph G on vertex set V = fv1; v2; : : : ; vng is a pair (V; E), where the edge set E is a subset of V × V. n is the order of G and jEj is the size of G. Computing Classification System 1998: G.2.2 Mathematics Subject Classification 2010: 05C30, 05C50 Key words and phrases: uniform hypergraph, independence number, lower bound DOI:10.2478/ausi-2014-0022 132 On vertex independence number of uniform hypergraphs 133 Let v 2 V and N(v) be the neighborhood of v, namely, the set of vertices x so that there is an edge which contains both v and x. Let U be a subset of V, then the subgraph of G induced by U is defined as a graph on vertex set U and edge set EU = f(u; v)ju 2 u and v 2 Ug.