A graph is $k$-total colourable if there is an assignment of $k$ different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product graphs are determined. In particular, a sufficient condition is given for direct products of bipartite graphs to have total chromatic number equal to its maximum degree plus one. Partial results towards the total chromatic number of $K_nimes K_m$ are also established.
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机译：一个图形是$ k $ -total Cootable如果将$ k $不同颜色的分配给图表的顶点和边缘，那么没有两个相邻的也没有入射元素接收相同的颜色。确定一些直接产品图的总色数。特别地，给予足够的条件，用于双链图的直接产物，以具有等于其最大程度的总色度。还建立了部分结果以K_N Times K_M $的总核数。
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