Good edge-labelings and graphs with girth at least five

2011/09/06 by Michel Bode, Bode, Michel, Babak Farzad +3
Mathematics · #05C78 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C78

paper · pdf · doi:10.48550/arxiv.1109.1125

23 pages + appendix

arxiv created 2012/07/27 · arxiv updated 2012/07/30

Abstract

A good edge-labeling of a graph [Araújo, Cohen, Giroire, Havet, Discrete Appl. Math., forthcoming] is an assignment of numbers to the edges such that for no pair of vertices, there exist two non-decreasing paths. In this paper, we study edge-labeling on graphs with girth at least 5. In particular we verify, under this additional hypothesis, a conjecture by Araújo et al. This conjecture states that if the average degree of G is less than 3 and G is minimal without an edge-labeling, then G ∈ C₃,K_2,3. (For the case when the girth is 4, we give a counterexample.)

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