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Choosability of bipartite graphs with maximum degree $Delta$

Stéphane Bessy 1 Frédéric Havet Jérôme Palaysi
1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Let G=(V(G), E(G)) be a graph. A list assignment is an assignment of a set L(v) of integers to every vertex v of G. An L-colouring is an application C from V(G) into the set of integers such that C(v)L(v) for all v V(G) and C(u)C(v) if u and v are joined by an edge. A (k,k')-list assignment of a bipartite graph G with bipartition (A,B) is a list assignment L such that |L(v)|= k if vA and |L(v)|= k' if vB. A bipartite graph is (k,k')-choosable if it admits an L-colouring for every (k.k')-list assignment L. In this paper, we study the (k,k')-choosability of graphs. Alon and Tarsi proved in an algebraic and non-constructive way, that every bipartite graph with maximum degree is (/2 +1, /2 +1)-choosable. In this paper, we give an alternative and constructive proof to this result. We conjecture that this result is sharp (i.e. there is a bipartite graph with maximum degree that is not (/2 , /2 +1)-choosable) and prove it for 5. Moreover, for a fixed , we show that given a bipartite graph with maximum degree and a (/2 , /2)+1)-list assignment L, it is NP-complete to decide if G is L-colourable. At last, we give upper bounds for the minimum size n_3() of a non (3,3)-choosable bipartite graph with maximum degree : n_3(5)846 and n_3(6)128.
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Stéphane Bessy, Frédéric Havet, Jérôme Palaysi. Choosability of bipartite graphs with maximum degree $Delta$. RR-4522, INRIA. 2002. ⟨inria-00072066⟩

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