The Alon–Tarsi number of planar graphs without cycles of lengths 4 and l

Abstract

This paper proves that if G is a planar graph without 4-cycles and l-cycles for some l ∈ {5, 6, 7}, then there exists a matching M such that AT (G − M ) ≤ 3. This implies that every planar graph without 4-cycles and l-cycles for some l ∈ {5, 6, 7} is 1-defective 3-paintable.

Decomposition of planar graphs with forbidden configurations

“…Indeed, stronger results were proved in [6,10]. The results concern two other graph parameters: The Alon-Tarsi number AT (G) of G and the paint number χ P (G) of G. The reader is referred to [6] for the definitions.…”

“…It was proved in [7] that there exists a planar graph G such that for any subgraph H of G of maximum degree 3, G − E(H) is not 3-choosable, and proved in [6] that every planar graph G has a matching M such that G − M is 4-choosable. For the second question, for l ∈ {5, 6, 7}, it was shown in [10] every graph G ∈ G 4,l has a matching M such that G − M is 3-choosable.…”

“…We just note here that for any graph G, ch(G) ≤ χ P (G) ≤ AT (G), and the differences χ P (G) − ch(G) and AT (G) − χ P (G) can be arbitrarily large. It was proved in [6] that every planar graph G has a matching M such that AT (G − M ) ≤ 4, and proved in [10] that for l ∈ {5, 6, 7}, every graph G ∈ G 4,l has a matching M such that AT (G − M ) ≤ 3.…”

“…In this paper, we consider further strengthening of the results concerning graphs in G 4,l for l ∈ {5, 6, 7, 8, 9}. (Note that the result in [10] does not cover the cases for l = 8 and 9). We strengthen the above results in two aspects: a larger class of graphs with a stronger property.…”

“…The following four lemmas first appeared in [10]. Even though the hypotheses of the theorems are different and some definitions are not the same, but the proofs for the following four lemmas are still true.…”