On minimum vertex cover of generalized Petersen graphs

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 $k$.

💡 Research Summary

The paper investigates the minimum vertex‑cover problem on the family of generalized Petersen graphs (P(n,k)). A generalized Petersen graph is defined for natural numbers (n) and (k) with (n>2k); its vertex set consists of two concentric cycles ({u_i}{i=1}^{n}) and ({v_i}{i=1}^{n}), and its edge set contains the “outer‑rim” edges (u_i u_{i+1}), the “spokes” (u_i v_i), and the “inner‑star” edges (v_i v_{i+k}) (indices taken modulo (n)). Because each vertex has degree three, the graph is 3‑regular and highly symmetric, which makes the vertex‑cover problem both challenging and amenable to structural analysis.

The authors first give a structural characterization of minimum vertex covers (MVCs). By exploiting the rotational symmetry of (P(n,k)), they show that any MVC must belong to one of two basic patterns. In the “outer‑centric” pattern most of the outer vertices (u_i) are selected, and a carefully chosen subset of inner vertices (v_i) is added to cover the remaining inner‑star edges. In the “inner‑cross” pattern the opposite occurs: a set of inner vertices is taken as the backbone, while outer vertices are added only where necessary. This dichotomy reduces the search space dramatically and provides a clear combinatorial description of all optimal covers.

Next, the paper derives lower bounds for the cover number (\beta(P(n,k))). The trivial bound (\beta \ge |E|/\Delta = n) follows from the fact that each vertex can cover at most three edges. A stronger bound is obtained by observing that the inner‑star edges form a collection of disjoint (k)-step chords. To cover each such chord at least one endpoint must be chosen, which yields \