A paired comparison digraph D is a weighted digraph in which the sum of the weights of arcs, if any, joining two vertices exactly one. A one-to-one mapping from V (D) onto {1, 2, . . . , |V (D)|} is called a ranking ofD, and a ranking α ofD is optimal if the backward length of α is minimum. We say that D is r-partite if V (D) can be partitioned into V1 ∪ · · · ∪ Vr so that every arc of D joinin...

Transversals in r-partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f -bounded transversals (or f -BT), that is, transversals whose connected components have order at most f . In some sense we search for the the sparsest f -BT-free graphs. We obtain estimates on the smallest maximum degree that 3-partite a...

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...

Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the...

Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ Here consider natural off-diagonal variant $r(B_{cn} B_n^{(k)})$ for fixed $c \in (0,1]$ more general setting, show that an interesting dichotomy emerges: very small $c$ , simple $k$ -partite construction ...

In this paper we extend the study of bipartite graphs with the induced ε-density property introduced by Frankl, Rödl, and the author. For a given kpartite k-uniform hypergraph G we say that a k-partite k-uniform hypergraph R = (W1, . . . ,Wk,F) has the induced ε-density property if every subhypergraph ofR with at least ε|F| edges contains a copy of G which is an induced subhypergraph of R. We s...

We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r − 4 for all but finitely many r. This answers a question of Abu-Khazneh, Barát, Pokrovskiy and Szabó, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r.

For a graph G, a partiteness k ≥ 2 and a number of colours c, we define the multipartite Ramsey number r k (G) as the minimum value m such that, given any colouring using c colours of the edges of the complete balanced k-partite graph with m vertices in each partite set, there must exist a monochromatic copy of G. We show that the question of the existence of r k (G) is tied up with what monoch...