This paper proves that every planar graph having no cycle of length 4, 7, or 9 is DP-3-colorable.
We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.
A digraph is called ubiquitous if every digraph that contains vertex-disjoint copies of for every also contains infinitely many vertex-disjoint copies of . We characterise which digraphs with rays as underlying undirected graphs are ubiquitous.
We say that a signed graph is -critical if it is not -colorable but every one of its proper subgraphs is -colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3-critical signed graph on vertices has at least edges, and that this bound is asymptotically tight. It follows that every signed planar or projective-planar graph of girth at least 6 is (circular) 3-colorable, and for the projective-planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph , which is the positive triangle augmented with a negative loop on each vertex.
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