Total Chromatic Number Of Graph Research Articles

Total chromatic number of graphs of high degree, II

AbstractWe prove Theorem 1: suppose G is a simple graph of order n having Δ(G) = n − k where k ≥ 5 and n ≥ max (13, 3k −3). If G contains an independent set of k − 3 vertices, then the TCC (Total Colouring Conjecture) is true. Applying Theorem 1, we also prove that the TCC is true for any simple graph G of order n having Δ(G) = n −5. The latter result together with some earlier results confirm that the TCC is true for all simple graphs whose maximum degree is at most four and for all simple graphs of order n having maximum degree at least n − 5.

The Total Chromatic Number of Graphs of High Minimum Degree

If G is a simple graph with minimum degree (G) satisfying (G) f(|V(G|+1) the total chromatic number conjecture holds; moreover if (G) |V(G| then T(G) (G)+3. Also if G has odd order and is regular with d{G) 7|(G)| then a necessary and sufficient condition for T(G) = (G)+1 is given.

Total chromatic number of graphs of high degree

AbstractUsing a new proof technique of the first author (by adding a new vertex to a graph and creating a total colouring of the old graph from an edge colouring of the new graph), we prove that the TCC (Total Colouring Conjecture) is true for any graph G of order n having maximum degree at least n - 4. These results together with some earlier results of M. Rosenfeld and N. Vijayaditya (who proved that the TCC is true for graphs having maximum degree at most 3), and A. V. Kostochka (who proved that the TCC is true for graphs having maximum degree 4) confirm that the TCC is true for graphs whose maximum degree is either very small or very big.

Total Colourings of Graphs

Basic terminology and introduction.- Some basic results.- Complete r-partite graphs.- Graphs of low degree.- Graphs of high degree.- Classification of type 1 and type 2 graphs.- Total chromatic number of planar graphs.- Some upper bounds for the total chromatic number of graphs.- Concluding remarks.