Abstract Generalizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity . In particular, together result due to Nash‐Williams Tutte, this implies 7‐edge‐connected bipartite whose edge set decomposes into two edge‐disjoint trees. We best possible as it does not hold for infinitely many 6‐edge‐connected gra...

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...

In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is largest minimum degree $\delta(G)$ among all $r$-partite graphs $G$ parts of size $n$ which do not contain a copy $K_{t+1}$? The $r=t+1$ case has attracted lot attention was fully resolved by Haxell Szab\'{o}, Szab\'{o} Tardos in 2006. this paper we investi...

A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying ${e(G) \gg v(G)^{1+1/k}}$ must contain $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies $C_{2k}$. Recently, Morris and Saxton established a balanced version Simonovits' theorem, showing such has $C_{2k}$, which are `uniformly distributed' over edges $G$. Moreover, they used this result to obtain sharp b...

Let q be a prime power and r = 0, 1 . . . , q − 3. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q− r)-regular bipartite graphs of girth 6 and q − rq− 1 vertices in each partite set. Moreover, in this work two La...

The complete multipartite graph with d partite sets, each containing m vertices, will be denoted by K(d; m). If all of the partite sets are not of the same cardinality, the graph K(mi, m2, . . . . md) is the complete d-partite graph with m, vertices in each partite sets, 1 <i<d. The special case where mi= 1 for every i, yields the complete graph on d vertices and is denoted by Kd. The join of g...

Extending the notion of (random) k-out graphs, we consider when the k-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each r there is a k = k(r) such that the k-out r-uniform hypergraph on n vertices has a perfect fractional matching with high probability (i.e., with probability tending to 1 as n → ∞) and prove an analogous result for r-uniform r-...

In this note, we show how optimal or approximate weighted independent sets in k-partite graphs may yield to a good approximate weighted independent set in general graphs. Precisely, optimal solutions in bipartite graphs do yield to a 2 ∆(G) approximation and, more generally, ρ-approximate solutions in k-partite graphs yield to a k ∆(G)ρ-approximation in general graphs.

Let H be a 3-partite 3-uniform hypergraph with each partition class of size n, that is, a 3-uniform hypergraph such that every edge intersects every partition class in exactly one vertex. We determine the Dirac-type vertex degree thresholds for perfect matchings in 3-partite 3-uniform hypergraphs.