Neighbour-connectivity in regular graphs

CONNECTIVITY OF RANDOM k-NEAREST-NEIGHBOUR GRAPHS

LetP be a Poisson process of intensity one in a squareSn of arean. We construct a random geometric graph Gn,k by joining each point of P to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn,k is connected tends to 1 as n → ∞. They conject...

Matching and edge-connectivity in regular graphs

Henning and Yeo proved a lower bound for the minimum size of a maximum matching in a connected k-regular graphs with n vertices; it is sharp infinitely often. In an earlier paper, we characterized when equality holds. In this paper, we prove a lower bound for the minimum size of a maximum matching in an l-edge-connected k-regular graph with n vertices, for l ≥ 2 and k ≥ 4. Again it is sharp for...

Neighbour-Distinguishing Edge Colourings of Random Regular Graphs

A proper edge colouring of a graph is neighbour-distinguishing if for all pairs of adjacent vertices v, w the set of colours appearing on the edges incident with v is not equal to the set of colours appearing on the edges incident with w. Let ndi(G) be the least number of colours required for a proper neighbour-distinguishing edge colouring of G. We prove that for d ≥ 4, a random d-regular grap...

A Note on Neighbour-Distinguishing Regular Graphs Total-Weighting

We investigate the following modification of a problem posed by Karoński, Luczak and Thomason [J. Combin. Theory, Ser. B 91 (2004) 151-157]. Let us assign positive integers to the edges and vertices of a simple graph G. As a result we obtain a vertex-colouring of G by sums of weights assigned to the vertex and its adjacent edges. Can we obtain a proper colouring using only weights 1 and 2 for a...

On the Connectivity of Random k-th Nearest Neighbour Graphs