78 research outputs found
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Some colouring problems for Paley graphs
The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved
The 1-2-3 Conjecture for Hypergraphs
A weighting of the edges of a hypergraph is called vertex-coloring if the
weighted degrees of the vertices yield a proper coloring of the graph, i.e.,
every edge contains at least two vertices with different weighted degrees. In
this paper we show that such a weighting is possible from the weight set
{1,2,...,r+1} for all hypergraphs with maximum edge size r>3 and not containing
edges solely consisting of identical vertices. The number r+1 is best possible
for this statement.
Further, the weight set {1,2,3,4,5} is sufficient for all hypergraphs with
maximum edge size 3, up to some trivial exceptions.Comment: 12 page
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