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Pré-Publication, Document De Travail Année : 2019

On {a,b}-edge-weightings of bipartite graphs with odd a,b

Résumé

For any $S⊂ℤ$ we say that a graph G has the S-property if there exists an S-edge-weighting $w:E(G)→S$ such that for any pair of adjacent vertices u,v we have $Σ_{e∈E(v)} w(e) ≠ Σ_{e∈E(u)} w(e)$, where E(v) and E(u) are the sets of edges incident to v and u respectively. This work focuses on {a,a+2}-edge-weightings where $a∈ℤ$ is odd. We show that a 2-connected bipartite graph has the {a,a+2}-property if and only if it is not a so-called odd multi-cactus. In the case of trees, we show that only one case is pathological. That is, we show that all trees have the {a,a+2}-property for odd $a≠−1$, while there is an easy characterization of trees without the {−1,1}-property.
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Dates et versions

hal-01988399 , version 1 (21-01-2019)
hal-01988399 , version 2 (28-08-2019)

Identifiants

  • HAL Id : hal-01988399 , version 1

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Julien Bensmail, Fionn Mc Inerney, Kasper Lyngsie. On {a,b}-edge-weightings of bipartite graphs with odd a,b. 2019. ⟨hal-01988399v1⟩
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