Let α(G) denote the independence number of a graph G and let G ×H be the direct product of graphs G and H. Set α(G ×H) = max{α(G) · |H|, α(H) · |G|}. If G is a path or a cycle and H is a path or a cycle then α(G×H) = α(G×H). Moreover, this equality holds also in the case when G is a bipartite graph with a perfect matching and H is a traceable graph. However, for any graph G with at least one ed...

An even cycle decomposition of a graph is a partition of its edge into even cycles. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5 is not the only graph in this class without such a decomposition. Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-c...

The best worst case guarantee algorithm to see if a graph has a Hamiltonian cycle, a closed tourvisiting every vertex exactly once, for a long time was based on dynamic programming over allthe vertex subsets of the graph. In this talk we will show some algebraic techniques that can beused to see if a graph has a Hamiltonian cycle much faster. These techniques utilize sums over<l...

SolC. P. and T. Zaslavsky, The covering radius of the cycle code of a graph, Discrete Applied Mathematics 45 (I 993) 63-70. The cycle code of a graph is the binary linear span of the characteristic vectors of circuits. We exploit a connection between the covering radius of this code and minimum T-joins. We obtain a lower bound on the covering radius which is met with equality when the graph is ...