Abstract
We restate theorems of Hutchinson [5] on list-colouring extendability for outerplanar graphs in terms of non-vanishing monomials in a graph polynomial, which yields an Alon-Tarsi equivalent for her work. This allows to simplify her proofs as well as obtain more general results.
Highlights
In his famous paper [8] Thomassen proved that every planar graph is 5-choosable
In this paper we provide a graph polynomial analogue to the result of Hutchinson, obtaining a characterisation of polynomial extendability for outerplanar graphs, which may be presented in the form of the following theorem
The three theorems above can be combined with Theorem 3.7 to obtain a general characterisation of (i, j)-extendability of outerplanar graphs
Summary
In his famous paper [8] Thomassen proved that every planar graph is 5-choosable. to proceed with an inductive argument, he proved the following stronger result. One, mentioned in [4], is the so called chain of diamonds, where the diamond is understood as K4 minus an edge It is obviously a 2-connected outerplane near triangulation, and the two non-neighbouring vertices are always of the same colour. The main tool connecting graph polynomials with list colourings is Combinatorial Nullstellensatz [1] It implies that for every non-vanishing monomial of P (G), if we assign to each vertex of G a list of length greater than the exponent of corresponding variable in that monomial, such list assignment admits a proper colouring. For background in graph theory see [9]
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