Abstract:  A K_⁴-homcomorph is a subdivision of the complete graph K_⁴.Such a homcomorph is denoted by K_⁴ (α,β,γ,δ,ε,η)if the six edges of K_⁴ are replaced by the six paths of length a，α,β,γ,δ,ε,η，
respectively. Since S. Kahn proposed the problem of chromaticity of K_⁴-homcomorphs, man mathematicians have been working on this problem and many beautiful results arc obtained. For example
the study of the chromaticity of K_⁴-homcomorphs with at least 2*-paths of length 1 has been fulfilled.So far,the study of the chromaticity of K_⁴-homcomorphs remains an active area of research. However,thc
further study of the chromaticity of K_⁴,-homcomorphs is quite complicated. So，in 2001，the author proposed a new classifying method in which the girth is chosen as the character of a graph. By this
method，some new results arc obtained. For instance, the study of the chromaticity of K_^4- homcomorphs which have girth less than 6 has been fulfilled. In this thesis, we fulfill the study of the chromaticity of
K_⁴,-homcomorphs which have girth 7. By using the way of classifying girth of graph,and considering the characteristic of chromatic polynomial，we find out some graphs with same chromatic polynomials but not
isomorphic to each other. Although the chromatic polynomial is a very powerful tool of describing the character of graph, for huge amount of complicate graphs with very few features and clues，it is almost
impossible to solve problems of chromaticity of graphs if we use chromatic polynomial simply. So, when study the chromaticity of K_⁴-homcomorphs,we adopt lots of technique to analyze the coefficients and the
powers of chromatic polynomials and so fulfilled the study of the chromaticity of K_⁴-homcomorphs which have firth 7.