Sharif Digital Repository

In this thesis, we study some bounds for the vertex chromatic num- ber and edge chromatic number of a graph. One of the most fa- mous theorems on graph colorings is Brooks’ Theorem, which asserts that every connected graph with maximum degree ∆(G) is ∆(G)- colorable unless G is an odd cycle or a complete graph. The following result has been proved: If every vertex of a graph G lies on at most k odd cycles for some nonnegative integer k, then χ(G) 1+√8k+9 . We recall from Vizing’s Theorem that the edge chromatic number of any graph must be equal either to ∆(G) or ∆(G) + 1. In this thesis, families of graphs that are Class 1 or Class 2 will be introduced.