On minimum vertex cover of generalized Petersen graphs

Abstract

For natural numbers $n$ and $k$ ($n > 2k$), a generalized Petersen graph $P(n,k)$, is defined by vertex set $\lbrace u_i,v_i\rbrace$ and edge set $\lbrace u_iu_{i+1},u_iv_i,v_iv_{i+k}\rbrace$; where $i = 1,2,\dots,n$ and subscripts are reduced modulo $n$. Here first, we characterize minimum vertex covers in generalized Petersen graphs. Second, we present a lower bound and some upper bounds for $\beta(P(n,k))$, the size of minimum vertex cover of $P(n,k)$. Third, in some cases, we determine the exact values of $\beta(P(n,k))$. Our conjecture is that $\beta(P(n,k)) \le n + \lceil\frac{n}{5}\rceil$, for all $n$ and $k$.