Silver Block Intersection Graphs of Steiner 2-Designs

Abstract

For a block design ${\mathcal{D}}$ , a series of block intersection graphs G _i , or i-BIG(${\mathcal{D}}$), i = 0, . . . ,k is defined in which the vertices are the blocks of ${\mathcal{D}}$ , with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-coloring of G. A vertex x in G is said to be rainbow with respect to c if every color appears in the closed neighborhood ${N[x] = N(x) \cup \{x\}}$ . Given an α-set I of G, a coloring c is said to be silver with respect to I if every ${x\in I}$ is rainbow with respect to c. We say G is silver if it admits a silver coloring with respect to some I. Finding silver graphs is of interest, for a motivation and progress in silver graphs see Ghebleh et al. (Graphs Combin 24(5):429–442, 2008) and Mahdian and Mahmoodian (Bull Inst Combin Appl 28:48–54, 2000). We investigate conditions for 0-BIG(${\mathcal{D}}$) and 1-BIG(${\mathcal{D}}$) of Steiner 2-designs ${{\mathcal{D}}=S(2,k,v)}$ to be silver.