Block Circulant Graphs and the Graphs of Critical Pairs of a Crown

TL;DR

This paper establishes a connection between block circulant graphs with symmetric growth patterns and the graphs of critical pairs of generalized crowns, providing bounds on their chromatic number.

Contribution

It introduces a natural bijection between a specific family of block circulant graphs and graphs of critical pairs of generalized crowns, offering new insights into their structure and coloring bounds.

Findings

Established a bijection between block circulant graphs and generalized crown graphs.

Provided an upper bound on the chromatic number for this family of graphs.

Characterized the structure of graphs with symmetric growth patterns in their block circulant matrices.

Abstract

In this paper, we provide a natural bijection between a special family of block circulant graphs and the graphs of critical pairs of the posets known as generalized crowns. In particular, every graph in this family of block circulant graphs we investigate has a generating block row that follows a symmetric growth pattern of the all ones matrix. The natural bijection provides an upper bound on the chromatic number for this infinite family of graphs.