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The 1-2-3 Conjecture, posed in 2004 by Karoński, Luczak, and Thomason, states that one may weight the edges of any connected graph on at least 3 vertices from the set {1, 2, 3} (call the weight function w) so that the function f(v) = ∑ u∈N(v) w(uv) is a proper vertex colouring. This paper presents the current state of research on the 1-2-3 Conjecture and(More)
An edge-weighting vertex colouring of a graph is an edge-weight assignment such that the accumulated weights at the vertices yield a proper vertex colouring. If such an assignment from a set S exists, we say the graph is S-weight colourable. We consider the S-weight colourability of digraphs by defining the accumulated weight at a vertex to be the sum of(More)
Karoński, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1, 2, 3} such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each edge’s weight may be chosen from an arbitrary(More)
In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle extendible; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on n vertices for any n ≥ 15. Furthermore, we show that there exist counterexamples where the ratio of the length(More)
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