Gábor Mészáros

ON PATH-PAIRABILITY IN THE CARTESIAN PRODUCT OF GRAPHS

Abstract We study the inheritance of path-pairability in the Cartesian product of graphs and prove additive and multiplicative inheritance patterns of path-pairability, depending on the number of vertices in the Cartesian product. We present path-pairable graph families that improve the known upper bound on the minimal maximum degree of a path-pairable graph. Further results and open questions about path-pairability are also presented.

On linkedness in the Cartesian product of graphs

We study linkedness of the Cartesian product of graphs and prove that the product of an a-linked and a b-linked graph is

Note on the diameter of path-pairable graphs

(a+b-1)

The path -pairability number of products of stars

(a+b-1)-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of products of paths and products of cycles.

An improved upper bound on the maximum degree of terminal-pairable complete graphs

A triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges. We study the set F n ? 0 , ( n 3 ) of possible number of frustrated triangles f ( G ) in a graph G on n vertices. We prove that about two thirds of the numbers in 0 , n 3 / 2 cannot appear in F n , and we characterise the graphs G with f ( G ) ? 0 , n 3 / 2 . More precisely, our main result is that, for each n ? 3 , F n contains two interlacing sequences 0 = a 0 ? b 0 ? a 1 ? b 1 ? ? ? a m ? b m ~ n 3 / 2 such that F n ? ( b t , a t + 1 ) = 0? for all t , where the gaps are | b t - a t + 1 | = ( n - 2 ) - t ( t + 1 ) and | a t - b t | = t ( t - 1 ) . Moreover, f ( G ) ? a t , b t if and only if G can be obtained from a complete bipartite graph by flipping exactly t edges/nonedges. On the other hand, we show, for all n sufficiently large, that if m ? f ( n ) , ( n 3 ) - f ( n ) , then m ? F n where f ( n ) is asymptotically best possible with f ( n ) ~ n 3 / 2 for n even and f ( n ) ~ 2 n 3 / 2 for n odd. Furthermore, we determine the graphs with the minimum number of frustrated triangles amongst those with n vertices and e ? n 2 / 4 edges.

On a conjecture of Gentner and Rautenbach

A graph on 2 k vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of at most 3. In this paper we present an infinite family of path-pairable graphs with diameter d ( G ) = O ( n ) where n denotes the number of vertices of the graph. We prove that our example is extremal up to a constant factor.

Terminal-Pairability in Complete Graphs

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

Note on terminal-pairability in complete grid graphs

Abstract The study of a graph theory model of certain telecommunications network problems lead to the concept of path-pairability, a variation of weak linkedness of graphs. A graph G is k-path-pairable if for any set of 2k distinct vertices, si, ti, 1 ≤ i ≤ k, there exist pairwise edge-disjoint si, ti-paths in G, for 1 ≤ i ≤ k. The path-pairability number is the largest k such that G is k-path-pairable. Cliques, stars, the Cartesian product of two cliques (of order at least three) are ‘fully pairable’; that is ⌊n/2⌋-pairable, where n is the order of the graph. Here we determine the path-pairability number of the Cartesian product of two stars.