Total-colouring of plane graphs with maximum degree nine
نویسندگان
چکیده
The central problem of the total-colourings is the Total-Colouring Conjecture, which asserts that every graph of maximum degree ∆ admits a (∆ + 2)-total-colouring. Similarly to edge-colourings—with Vizing’s edge-colouring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if ∆ ≥ 10 then every plane graph of maximum degree ∆ is (∆ + 1)-totally-colourable. On the other hand, such a statement does not hold if ∆ ≤ 3. We prove that every plane graph of maximum degree 9 can be 10-totally-coloured.
منابع مشابه
Edge-face coloring of plane graphs with maximum degree nine
An edge-face colouring of a plane graph with edge set E and face set F is a colouring of the elements of E ∪ F so that adjacent or incident elements receive different colours. Borodin [Simultaneous coloring of edges and faces of plane graphs, Discrete Math., 128(1-3):21–33, 1994] proved that every plane graph of maximum degree ∆ > 10 can be edge-face coloured with ∆ + 1 colours. We extend Borod...
متن کاملTotal-Coloring of Plane Graphs with Maximum Degree Nine
The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree ∆ admits a (∆+2)-total-coloring. Similar to edge-colorings—with Vizing’s edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if ∆ ≥ 10, then every plane graph of maximum degree ∆ is (∆ + 1)...
متن کاملTotal-Chromatic Number and Chromatic Index of Dually Chordal Graphs
A graph is dually chordal if it is the clique graph of a chordal graph. Alternatively, a graph is dually chordal if it admits a maximum neighbourhood order. This class generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs and interval graphs. We prove that Vizing's total-colour conjecture holds for dually chordal graphs. We describe a new heuristi...
متن کاملList edge-colouring and total colouring in graphs of low treewidth
We prove that the list chromatic index of a graph of maximum degree ∆ and treewidth ≤ √ 2∆ − 3 is ∆; and that the total chromatic number of a graph of maximum degree ∆ and treewidth ≤ ∆/3 + 1 is ∆ + 1. This improves results by Meeks and Scott.
متن کاملAn upper bound for total colouring of graphs
McDiarmid C.J.H. and A. Sanchez-Arroyo, An upper bound for total colouring of graphs, Discrete Mathematics 111 (1993) 3899392. We give an upper bound on the number of colours required to extend a given vertex colouring of a graph to a total colouring. This shows that for any simple graph there is a total colouring using at most :d + 3 colours, where A is the maximum vertex degree.
متن کامل