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We show that the Yao graph Y 4 in the L 2 metric is a spanner with stretch factor 8 √ 2(29+ 23 √ 2). Enroute to this, we also show that the Yao graph Y ∞ 4 in the L∞ metric is a plane spanner with stretch factor 8.

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)

- Ben Seamone
- ArXiv
- 2012

The 1-2-3 Conjecture, posed in 2004 by Karo´nski, 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)

- Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'Rourke, Ben Seamone, Michiel H. M. Smid +1 other
- ISAAC
- 2010

We show that the Yao graph Y 4 in the L 2 metric is a spanner with stretch factor 8(29+23 √ 2). Enroute to this, we also show that the Yao graph Y ∞ 4 in the L ∞ metric is a planar spanner with stretch factor 8.

Karo´nski, 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)

A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r-regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen.

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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