The complete multipartite graph Kn(2t) having n partite sets of size 2t, with n ≥ 6 and t ≥ 1, is shown to have a decomposition into gregarious 6-cycles, that is, the cycles which have at most one vertex from any particular partite set. Complete sets of differences of numbers in Zn are used to produce starter cycles and obtain other cycles by rotating the cycles around the n-gon of the partite ...

in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.

We strengthen the stability theorem of Erdős and Simonovits. Write Kr (s1, . . . , sr) for the complete r-partite graph with parts of size s1, . . . , sr and Tr (n) for the r-partite Turán graph of order n. Our main result is: For all r ≥ 2 and sufficiently small c > 0, ε > 0, for every graph G of order n with ⌈ (1− 1/r − ε)n2/2 ⌉ edges, one of the following conditions holds: (a) Kr+1 ( ⌊c lnn⌋...

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call (D)=max16 i6 n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1; V2; : : : ; Vn are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and t...

Let K+ r (s1, . . . , sr) be the complete r-partite graph with parts of size s1 ≥ 2, s2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε > 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ( ⌊c lnn⌋ , . . . , ⌊c lnn⌋ , ⌈ ...

The p-spectral radius of a uniform hypergraph G of order n is de…ned for every real number p 1 as (p) (G) = max jx1j + + jxnj = 1 r! X fi1;:::;irg2E(G) xi1 xir : It generalizes several hypergraph parameters, including the Lagrangian, the spectral radius, and the number of edges. The paper presents solutions to several extremal problems about the p-spectral radius of k-partite and k-chromatic hy...

The trace norm ‖G‖∗ of a graph G is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm ‖G‖∗ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note focuses on the maximum trace norm of r-partite graphs, which for r > 2 raises some unusual problems. It is shown that, if G is an r-partite graph of order n, ...

It is known that every hypercube Qn is a bipartite graph. Assume that n 2 and F is a subset of edges with |F | n− 2. We prove that there exists a hamiltonian path in Qn − F between any two vertices of different partite sets. Moreover, there exists a path of length 2n − 2 between any two vertices of the same partite set. Assume that n 3 and F is a subset of edges with |F | n− 3. We prove that th...

A digraph obtained by replacing each edge of a complete n-partite (n 2:: 2) graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete n-partite digraph or semicomplete multipartite digraph (abbreviated to SMD). In this paper we show the following result for a semicomplete multipartite digraph of order p with the partite sets VI, 112, ... , Vn. Let r...

Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D) : D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)| − 1 for any connected graph G, the present pape...