Triple intersection numbers for the Paley graphs

Intersection Numbers of Kirkman Triple Systems

A Steiner triple system of order v (briefly STS(v)) is a pair (X, B) where X is a v-set and B is a collection of 3-subset of X (called triple) such that every pair of distinct elements of X belongs to exactly one triple of B. A Kirkman triple system of order v (briefly KTS(v)) is a Steiner triple system of order v (X, B) together with a partition R of the set of triples B into subsets R1 , R2 ,...

Block-Intersection Graphs of Steiner Triple Systems

A Steiner triple system of order n is a collection of subsets of size three, taken from the n-element set {0, 1, ..., n−1}, such that every pair is contained in exactly one of the subsets. The subsets are called triples, and a block-intersection graph is constructed by having each triple correspond to a vertex. If two triples have a non-empty intersection, an edge is inserted between their vert...

Paley triple arrays

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

‎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 ver...

The watchman's walk of Steiner triple system block intersection graphs

A watchman’s walk in a graph G = (V,E) is a minimum closed dominating walk. In this paper, it is shown that the number of vertices in a watchman’s walk on the block intersection graph of a Steiner triple system is between v−3 4 and v−7 2 , for admissible v ≥ 15. Included are constructions to build a design that achieves the minimum bound for any admissible v.