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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.
Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.
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.
Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $chi(G)$, and the maximum is the so-cal
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 b
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $omega$ has chromatic number at most $3cdot 4^{omega-1}