The asymptotic covering density of generalized Petersen graphs

Remark on the paper "Minimum vertex covers in the generalized Petersen graphs P(n; 2)" by M. Behzad, P. Hatami, and E.S. MahmoodianDedicated to Tomaz Pisanski on the occasion of his 60th birthdayThe covering density of a graph G=(V,E) is delta(G)= beta(G)/|V|where beta(G), the covering number, is the minimum number of vertices that represent all edges of G. The asymptotic covering density of the generalized Petersen graph is determined

A Variation of Decomposition Under a Length Constraint

Introducing and initiating a study of a new variation of decomposition namely equiparity induced path decomposition of a graph which is defined to be a decomposition in which all the members are induced paths having same parity

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201