If G and H are vertex-transitive graphs, then the framing number fr(G,H) of G and H is defined as the minimum order of a graph every vertex of which belongs to an induced G and an induced H. This paper investigates fr(Cm, Cn) for m < n. We show first that fr(Cm, Cn) ≥ n+2 and determine when equality occurs. Thereafter we establish general lower and upper bounds which show that fr(Cm, Cn) is app...

For c ∈ (0, 1) let Pn(c) denote the set of n-vertex perfect graphs with density c and Cn(c) the set of n-vertex graphs without induced C5 and with density c. We show that log2 |Pn(c)|/ ( n 2 ) = log2 |Cn(c)|/ ( n 2 ) = h(c) + o(1) with h(c) = 12 if 1 4 ≤ c ≤ 3 4 and h(c) = 1 2H(|2c− 1|) otherwise, where H is the binary entropy function. Furthermore, we use this result to deduce that almost all ...

A kernel J of a digraph D is an independent set of vertices of D such that for every vertex w ∈ V (D)\J there exists an arc from w to a vertex in J . In this paper we have obtained results for the existence and nonexistence of kernels in Cartesian products of certain families of digraphs, and characterized T Cn, T Pn and Cm Cn which have kernels, where T is a tournament, and Pn and Cn are, resp...

In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G) γ(H) fixing one factor. For the complete bipartite graphs Km,n; we characterize all nontrivial solutions when m = 2, n ≥ 3 and prove the nonexistence of solutions when m, n ≥ 3. In addition, it is proved that the above equation has no nontrivial solution if H is one of the graphs obtained from Cn,...

Thewheel graph, denoted byWn+1, is the graph obtained from the circuit Cn with n vertices by adding a new vertex and joining it to every vertex of Cn. In this paper, the wheel graph Wn+1, except for W7, is proved to be determined by its Laplacian spectrum, and a graph cospectral with the wheel graphW7 is given. © 2009 Elsevier Ltd. All rights reserved.

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles o one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H , f(G H) f(G)f(H). Let Cm and Cn be cycles. We prove that f(Cm Cn) f(Cm)f(Cn) for all but a ni...