We present the tables of feasible parameters primitive 3 -class Q -polynomial association schemes and 4 - 5 -bipartite (on up to 2800 , 10000 50000 vertices, respectively), accompanied by a number nonexistence results for such obtained analysing triple intersection numbers putative open cases.

In this paper we will look at the relationship between the intersection number c2 and its diameter for a distance-regular graph. And also, we give some tools to show that a distance-regular graph with large c2 is bipartite, and a tool to show that if kD is too small then the distance-regular graph has to be antipodal.

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an ...

In this paper, we consider the intersection graph IΓ(Zn) of gamma sets in the total graph on Zn. We characterize the values of n for which IΓ(Zn) is complete, bipartite, cycle, chordal and planar. Further, we prove that IΓ(Zn) is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivit...

Let t(C) be the number of tangent pairs among a set C of n Jordan regions in the plane. Pach, Suk, and Treml [6] showed that if C consists of convex bodies and its intersection graph is bipartite then t(C) ≤ 4n−Θ(1), and, moreover, there are such sets that admit at least 3n−Θ( √ n) tangencies. We close this gap and generalize their result by proving that the correct bound is 3n−Θ(1), even when ...

let $g$ be a finite group with the identity $e$‎. ‎the subgroup intersection graph $gamma_{si}(g)$ of $g$ is the graph with vertex set $v(gamma_{si}(g)) = g-e$ and two distinct vertices $x$ and $y$ are adjacent in $gamma_{si}(g)$ if and only if $|leftlangle xrightrangle capleftlangle yrightrangle|>1$‎, ‎where $leftlangle xrightrangle $ is the cyclic subgroup of $g$ generated by $xin g$‎. ‎in th...

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an l × l grid minor is exponential in l. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order l in a graph G is a set of paths in G whose intersection graph is bipar...

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an l × l grid minor is exponential in l. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order l in a graph G is a set of paths in G whose intersection graph is bipar...

1 The MinRep problem The challenge of making k << |T | is highly non trivial. The difficulty is among other things the need for reduction for subset k-connectivity. The construction is essentially new and has only minor intersection with the one from [KKL]. Generally speaking, we must make the reduction local. For example in the sense that we mainly connect vertices at distance 2. We describe t...