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

Julien Bensmail 1 Fionn Mc Inerney 1 Kasper Lyngsie 2
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : 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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Julien Bensmail, Fionn Mc Inerney, Kasper Lyngsie. On {a,b}-edge-weightings of bipartite graphs with odd a,b. Discussiones Mathematicae Graph Theory, University of Zielona Góra, In press. ⟨hal-01988399v2⟩

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