We put the final piece into a puzzle first introduced by Bollobás, Erdős and Szemerédi in 1975. For arbitrary positive integers n and r we determine the largest integer ∆ = ∆(r, n), for which any r-partite graph with partite sets of size n and of maximum degree less than ∆ has an independent transversal. This value was known for all even r. Here we determine the value for odd r and find that ∆(...

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...

We investigate and quantify bipartite tripartite entanglement measures in two- three-flavor neutrino oscillations. The is analogous to the swapping resulting from a beam splitter quantum optics. calculate various measures, such as concurrence, negativity, three-tangle for three-neutrino system. significant result that monogamy inequality terms of negativity leads residual entanglement, implying...

Let μ (G) be the largest eigenvalue of a graph G and Tr (n) be the r-partite Turán graph of order n. We prove that if G is a graph of order n with μ (G) > μ (Tr (n)) , then G contains various large supergraphs of the complete graph of order r + 1, e.g., the complete r-partite graph with all parts of size log n with an edge added to the first part. We also give corresponding stability results.

We find the minimal number of links in an embedding of any complete k-partite graph on 7 vertices (including K 7 , which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete k-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with min...

Ryser’s conjecture postulates that, for r-partite hypergraphs, τ ≤ (r − 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960s, researchers have resolved it for special cases such as for intersecting hypergraphs where r ≤ 5. In this paper, we prove several results pertaining to matchings and coverings in r-partit...

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...