Geodesic
Digraphs are 2-weight choosable
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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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 the inbound weights minus the sum of the outbound weights. Bartnicki et al. showed that every digraph is $S$-weight colourable for any set $S$ of size $2$ and asked whether one could show the same result using an algebraic approach. Using the Combinatorial Nullstellensatz and a classical theorem of Schur, we provide such a solution.
DOI : 10.37236/508
Classification : 05C22, 05C15
Mots-clés : edge weighting vertex coloring, weight colorability, digraphs 
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     author = {Mahdad Khatirinejad and Reza Naserasr and Mike Newman and Ben Seamone and Brett Stevens},
     title = {Digraphs are 2-weight choosable},
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     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/508},
     zbl = {1205.05101},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/508/}
}
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Mahdad Khatirinejad; Reza Naserasr; Mike Newman; Ben Seamone; Brett Stevens. Digraphs are 2-weight choosable. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/508

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