This paper examines how close the chordal SLEκ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ∈ (0, 4), it is shown that if β > βκ := 1/(8/κ − 2), then the intersection of the SLEκ curve with the graph of the function y = x/(log x)β, x > e, is a.s. bounded, while it is a.s. unbounded if β = βκ. The critical SLE4 curve a.s. intersects the graph...

The problem of decomposing the block intersection graph of a Steiner triple system into triangles is considered. In the case when the block intersection graph has even degree, this is completely solved, while when the block intersection graph has odd degree, removal of some spanning subgraph of odd degree is necessary before the rest can be decomposed into triangles. In this case, some decompos...

An axis parallel d-dimensional box is the cartesian product R1 × R2 × · · · × Rd where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer d such that G can be represented as the intersection graph of a collection of d-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. Permutati...

Experimental results show that in large complex networks such as internet or biological networks, there is a tendency to connect elements which have a common neighbor. This tendency in theoretical random graph models is depicted by the asymptotically constant clustering coefficient. Moreover complex networks have power law degree distribution and small diameter (small world phenomena), thus the...

A uniform random intersection graph G(n,m, k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when m...

In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected g...

J.H. Koolen and J. Park proved a lower bound for the intersection number c2 of a distance-regular graph Γ. Moreover, they showed that a graph Γ, for which equality is attained in this bound, is a Terwilliger graph. We prove that Γ is the icosahedron, the Doro graph or the Conway–Smith graph if equality is attained and c2 > 2.