The Maximum Independent Set problem on circulant graphs Cn(a,b)

Abstract

A circulant graph $C_n(a,b)$ is a graph with $n$ vertices $\{v_0,\dots ,v_{n-1}\}$ such that each vertex $v_i$ is adjacent to vertices $v_{(i+a)modn}$ and $v_{(i+b)modn}$. In this paper, we investigate the Maximum Independent Set problem on 4-regular connected circulant graphs $C_n(a,b)$, with arbitrary $n$, $a$, and $b$. We approach the problem in an algebraic and combinatorial way based on an array representation of the graph. We prove that the cardinality of a maximum independent set is equal to $\frac{n-\left|W^{\ast }\right|}{2}$ where $\left|W^{\ast }\right|$ is the cardinality of a minimum subset of vertices that has a nonempty intersection with every odd cycle. The approach we propose shows that the Maximum Independent Set problem on any $C_n(a,b)$ can be solved in $O\left(n\right)$ time. As a consequence, covering number, odd cycle transversal, and fractional chromatic number are all linear time solvable on these graphs.