On the chromaticity of the 2-degree integral subgraph ofq-trees

A graphG is called to be a 2-degree integral subgraph of aq-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactlyq- 1 triangles. An added-vertexq-treeG with n vertices is obtained by taking two verticesu, v (u, v are not adjacent) in a q-treesT withn -1 vertices such that their intersection of neighborhoods ofu, v forms a complete graphK ^sub q^, and adding a new vertexx, new edgesxu, xv, xv ^sub 1^,xv ^sub 2^, ...,xv ^sub q- 4^, where {v ^sub 1^,v ^sub 2^,...,v ^sub q^-4} -K ^sub q^. In this paper we prove that a graphG with minimum degree not equal toq -3 and chromatic polynomialP(G;λ) = λ(λ - 1) ... (λ -q +2)(λ -q +1)^sup 3^(λ -q)^sup n- q- 2^ withn ≥ q + 2 has and only has 2-degree integral subgraph of q-tree withn vertices and added-vertex q-tree withn vertices.[PUBLICATION ABSTRACT]