Silver block intersection graphs of Steiner 2-designs

Abstract

For a block design $\mathcal{D}$, a series of {\sf block intersection graphs} $G_i$, or $i$-{\rm 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 a {\sf diagonal} of that graph. Let $G$ be $r$-regular and $c$ be a proper $(r+1)$-coloring of $G$. A vertex $x$ in $G$ is said to be {\sf rainbow} with respect to $c$ if every color appears in the closed neighborhood $N[x]=N(x)\cup \{x\}$. Given a diagonal $I$ 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 {\sf silver} if it admits a silver coloring with respect to some $I$. We investigate conditions for 0-{\rm BIG}($\mathcal{D}$) and 1-{\rm BIG}($\mathcal{D}$) of Steiner systems $\mathcal{D}=S(2,k,v)$ to be silver.