We prove that the Petersen colouring conjecture implies a conjecture of Markström saying that the line graph of every bridgeless cubic graph is decomposable into cycles of even length. In addition, we describe two infinite families of 4regular graphs: the first family consists of 3-connected graphs with no even cycle decomposition and the second one consists of 4-connected signed graphs with no...

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-conn...

A graph G is called even if O(G)=<, and G is called eulerian if G is even and connected. If G has a spanning eulerian subgraph, then G is called supereulerian, and we write G # SL. Tutte [19, 20] and Matthews [17] conjectured that if a 2-edge-connected graph G has no subgraph contractible to the Petersen graph, then G has a 3-colorable double cycle cover (i.e., a collection of three even subgra...

Abstract A multilevel distance labeling of the graph G is a function f = (V (G),E(G)) on V (G) of G into N ∪ {0} so that |f(u) − f(v)| ≥ diam(G) + 1− d(u, v) for all u, v ∈ V (G). The smallest span taken over all multilevel distance labeling of G is radio number rn(G) of G. In this paper, we completely determine the radio number rn(G) of G where G is the graph obtained from generalized Petersen...

Let G^s be a signed graph, where G = (V;E) is the underlying simple graph and s : E(G) to {+, -} is the sign function on E(G). In this paper, we obtain k-th signed spectral moment and k-th signed Laplacian spectral moment of Gs together with coefficients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.

2010 Mathematics Subject Classifications: Primary 05C22; Secondary 05C15, 05C25, 05C30 Abstract. Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illu...

We prove that every 3-regular graph with no circuit of length less than six has a subgraph isomorphic to a subdivision of the Petersen graph. 3 January 1997 Revised 17 March 2014 ∗ Research partially performed under a consulting agreement with Bellcore, and partially supported by DIMACS Center, Rutgers University, New Brunswick, New Jersey 08903, USA. 1 Partially supported by NSF under Grant No...

We prove that the number of 1-factorisations of a generalised Petersen graph of the type GP (3k, k) is equal to the kth Jacobsthal number J(k) if k is odd, and equal to 4J(k), when k is even. Moreover, we verify the list colouring conjecture for GP (3k, k).