Two hamiltonian paths P1 = 〈u1,u2, . . . ,un(G)〉 and P2 = 〈v1,v2, . . . ,vn(G)〉 of G from u to v are independent if u = u1 = v1, v = vn(G) = un(G), and vi = ui for every 1 < i < n(G). A set of hamiltonian paths, {P1,P2, . . . ,Pk }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v . A bipartite graph G is hamiltonian laceable if there ...

In this paper we study multipartite Ramsey numbers for odd cycles. Our main result is the proof of a conjecture of Gyárfás, Sárközy and Schelp [12]. Precisely, let n ≥ 5 be an arbitrary positive odd integer; then in any two-coloring of the edges of the complete 5-partite graph K(n−1)/2,(n−1)/2,(n−1)/2,(n−1)/2,1 there is a monochromatic cycle of length n. keywords: cycles, Ramsey number, Regular...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

For a given r-uniform hypergraph F we study the largest blow-up of F which can be guaranteed in every large r-uniform hypergraph with many copies of F . For graphs this problem was addressed by Nikiforov, who proved that every n-vertex graph that contains Ω(n`) copies of the complete graph K` must contain a complete `-partite graph with Ω(logn) vertices in each class. We give another proof of N...

Let Kk,n be a complete k-partite graph of order n and let K k,n be a generalized complete k-partite graph of order n spanned by the fan set F = {Fn1 , Fn2 , · · · , Fnk}, where n = {n1, n2, · · · , nk} and n = n1 + n2 + · · ·+ nk for 1 6 k 6 n. In this paper, we get the number of spanning trees in Kk,n to be t(Kk,n) = n k−2 k ∏ i=1 (n− ni)i. and the number of spanning trees in K k,n to be t(K k...

The honeycomb rectangular torus is an attractive alternative to existing networks such as mesh-connected networks in parallel and distributed applications because of its low network cost and well-structured connectivity. Assume that m and n are positive even integers with n P 4. It is known that every honeycomb rectangular torus HReTðm; nÞ is a 3-regular bipartite graph. We prove that in any HR...

In this paper, it is shown that the graph obtained by overlapping the cycle and the complete tripartite graph at an edge is uniquely determined by its chromatic polynomial. ) 3 ( ≥ m Cm 2 , 2 , 2 K Let G be a finite graph with neither loops nor multiple edges and let ) ; ( λ G P denote its chromatic polynomial. Then G is said to be chromatically unique if ) ; ( ) ; ( λ λ G Y P = implies that Y ...

Let R be a commutative ring and G(R) be a graph with vertices as proper andnon-trivial ideals of R. Two distinct vertices I and J are said to be adjacentif and only if I + J = R. In this paper we study a graph constructed froma subgraph G(R)Δ(R) of G(R) which consists of all ideals I of R such thatI Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper westudy about the relation b...

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let Tn,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that Tn,r is the only extremal graph for complete graph Kr+1. Erdős et al. (1995) determined the ...

In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, . . . , mr&minus1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs are co...