Silver block intersection graphs of Steiner 2-designs

TL;DR

This paper studies the properties of block intersection graphs derived from Steiner 2-designs, focusing on conditions under which these graphs admit special colorings called silver colorings, with implications for graph symmetry and design theory.

Contribution

It introduces the concept of silver colorings in block intersection graphs of Steiner 2-designs and characterizes when these graphs are silver, expanding understanding of graph colorings in combinatorial designs.

Findings

Conditions for 0-BIG to be silver are established.

Conditions for 1-BIG to be silver are characterized.

New connections between Steiner systems and graph colorings are identified.

Abstract

For a block design $\cal{D}$, a series of {\sf block intersection graphs} $G_i$, or $i$-{\rm BIG}($\cal{D}$), $i=0, ..., k$ is defined in which the vertices are the blocks of $\cal{D}$, 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}($\cal{D}$) and 1-{\rm BIG}($\cal{D}$) of Steiner systems ${\cal{D}}=S(2,k,v)$ to be silver.