## Silver block intersection graphs

#### Citation

Ahadi, A., Besharati, N., Mahmoodian, E. S. ve Mortezaeefar, M. (2009). Silver block intersection graphs. International Conference on Mathematical Sciences, Maltepe Üniversitesi. s. 29-30.#### Abstract

Any maximum independent set of a graph is called a diagonal of that graph. Let c be a proper (r + 1)-coloring of an
r-regular graph G. A vertex in G is said to be rainbow with respect to c if every color appears in the closed neighborhood
N[x] = N(x) [ fxg. Given a diagonal I of G, the coloring c is said to be silver with respect to I if every x 2 I is rainbow
with respect to c. We say G is silver if it admits a silver coloring with respect to some I. In [1] the following problem is
asked: Find classes of silver r-regular graphs G.
Here we study the class of block intersection graphs of Steiner triple systems, STS(v). Given a design D, a series of
block intersection graphs G1, or i-BIG, i = 0,... k; can be 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 points. Let D be an STS(v), G2 and G3 are
empty graphs, so we consider only G0 and G1. G0 is a strongly regular graph SRG(b; b ¡ 3r + 2; b ¡ 6r + 13; b ¡ 5r + 8),
and G1 is an SRG(b; 3(r ¡ 1); r + 2; 9): The aim of this talk is to characterize G0, and G1 for being silver.
We show that:
² For v = 7 and 9, G0 and G1 both are silver.
² For any STS(13) or STS(15) non of G0 or G1 are silver.
² Let D be an affine plane of order n. Then both 0-BIG(D) and 1-BIG(D) are silver.
² For each w, where 9jIw, we construct a Steiner triple system D = STS(w) for which, the 1-BIG(D) is silver.
² For any v > 9, 0-BIG(STS(v)) is not silver.
² If 9 - v and an STS(v) which has v3
parallel class, then G1 = 1-BIG(STS(v)) is not silver.
² If 9 - (v ¡ 1) and an STS(v) which contains v¡1
3 parallel class, then G1 = 1-BIG(STS(v)) is not silver.

#### Source

International Conference on Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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