Silver Block Intersection Graphs of Steiner 2-Designs

For a block design D, a series of block intersection graphs Gi, or i-BIG(D), i = 0, . . . , k is defined in which the vertices are the blocks of 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)∪{x}. Given an α-set I of G, a coloring c is said to be silver with respect to I if every x ∈ 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 [7] and [15]. We investigate conditions for 0-BIG(D) and 1-BIG(D) of Steiner 2-designs D = S(2, k, v) to be silver. keywords: Silver coloring, Block intersection graph, Steiner 2-design, and Steiner triple system Subject class: 05C15, 05B05, 05B07, and 05C69