A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given.
We consider face-to-face partitions of bounded polytopes into convex polytopes in d for arbitrary d ≥ 1 and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed d+ 1. Partitions of polyhedra in 3 into pentahedra and hexahedra are 5- and 6-colourable, respectively. We show that the above numbers are attainable, i.e., in general, they cannot be reduced.
Kragujevac (M. L. Kragujevac: On the Laplacian energy of a graph, Czech. Math. J. {\it 56}({\it 131}) (2006), 1207--1213) gave the definition of Laplacian energy of a graph $G$ and proved $LE(G)\geq 6n-8$; equality holds if and only if $G=P_n$. In this paper we consider the relation between the Laplacian energy and the chromatic number of a graph $G$ and give an upper bound for the Laplacian energy on a connected graph.