Some lower bounds for the $L$-intersection number of graphs

Document Type: Research Paper

Authors

1 Department of Mathematical Sciences‎, ‎Isfahan University of Technology‎, ‎84156-83111‎, ‎Isfahan‎, ‎Iran.

2 Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

Abstract

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v \subseteq \{1,\dots‎, ‎l\}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u \cap A_v|\in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the vertices in different parts‎. ‎In this paper‎, ‎some lower bounds for the (bipartite) $L$-intersection number of a graph for various types $L$ in terms of the minimum rank of graph are obtained‎. ‎To achieve the main results we employ the inclusion matrices of set systems and show that how the linear algebra techniques give elegant proof and stronger results in some cases.

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Main Subjects


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Volume 43, Issue 1
January and February 2017
Pages 69-78
  • Receive Date: 11 January 2015
  • Revise Date: 10 October 2015
  • Accept Date: 10 October 2015