The total-chromatic number of some families of snarks
نویسندگان
چکیده
The total chromatic number χ T (G) is the least number of colours needed to colour the vertices and edges of a graph G, such that no incident or adjacent elements (vertices or edges) receive the same colour. It is known that the problem of determining the total chromatic number is NP-hard and it remains NP-hard even for cubic bipartite graphs. Snarks are simple connected bridgeless cubic graphs which are not 3-edge colourable. In this paper, we show that the total chromatic number is 4 for three infinite families of snarks, namely, the Flower Snarks, the Goldberg Snarks and the Twisted Goldberg Snarks. This result reinforces the conjecture that all snarks are type 1. Moreover, we give recursive procedures to construct 4-total colourings in each case.
منابع مشابه
Diana Sasaki Simone Dantas Celina
Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Conjecture. In 2003, Cavicchioli et al. proved that for snarks with less than 30 vertices, the total chromatic number is 4, and proposed the problem of finding (if any) the smallest snark which is not 4-total colorable. Several families of snarks have had their total...
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A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT (G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and c...
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عنوان ژورنال:
- Discrete Mathematics
دوره 311 شماره
صفحات -
تاریخ انتشار 2011