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Register a new userA Unified Approach to Distance-Two Colouring of Graphs on Surfaces
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In this paper we introduce the notion of $Sigma$-colouring of a graph $G$: For given subsets $Sigma(v)$ of neighbours of $v$, for every $vin V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $Sigma(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptot
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This paper disproves a conjecture of Wang, Wu, Yan and Xie, and answers in negative a question in Dvorak, Pekarek and Sereni. In return, we pose five open problems.
We present a method which provides a unified framework for most stability theorems that have been proved in graph and hypergraph theory. Our main result reduces stability for a large class of hypergraph problems to the simpler question of checking that a hypergraph $mathcal H$ with large minimum degree that omits the forbidden structures is vertex-extendable. This means that if $v$ is a vertex of $mathcal H$ and ${mathcal H} -v$ is a subgraph of the extremal configuration(s), then $mathcal H$ is also a subgraph of the extremal configuration(s). In many cases vertex-extendability is quite easy to verify. We illustrate our approach by giving new short proofs of hypergraph stability results of Pikhurko, Hefetz-Keevash, Brandt-Irwin-Jiang, Bene Watts-Norin-Yepremyan and others. Since our method always yields minimum degree stability, which is the strongest form of stability, in some of these cases our stability results are stronger than what was known earlier. Along the way, we clarify the different notions of stability that have been previously studied.
A (not necessarily proper) vertex colouring of a graph has clustering $c$ if every monochromatic component has at most $c$ vertices. We prove that planar graphs with maximum degree $Delta$ are 3-colourable with clustering $O(Delta^2)$. The previous best bound was $O(Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $Delta$ that exclude a fixed minor are 3-colourable with clustering $O(Delta^5)$. The best previous bound for this result was exponential in $Delta$.
In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.
A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs. In another paper, two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three-edge-colourable. The proof method is a variant on the proof of the four-colour theorem.
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