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We use $P_t$ and $C_t$ to denote a path and a cycle on t vertices,respectively. A diamond consists of two triangles that share exactly one edge,a kite is a graph obtained from a diamond by adding a new vertex adjacent to avertex of degree 2 of the diamond, a paraglider is the graph that consists of a$C_4$ plus a vertex adjacent to three vertices of the $C_4$, a paw is a graphobtained from a triangle by adding a pendant edge. A comparable pair $(u, v)$consists of two nonadjacent vertices $u$ and $v$ such that $N(u)\subseteq N(v)$or $N(v)\subseteq N(u)$. A universal clique is a clique $K$ such that $xy \inE(G)$ for any two vertices $x \in K$ and $y\in V (G)\setminus K$. A blowup of agraph H is a graph obtained by substituting a stable set for each vertex, andcorrespondingly replacing each edge by a complete bipartite graph. We provethat 1) there is a unique connected imperfect $(P_7, C_5$, kite,paraglider)-free graph G with \delta(G) \geq \omega(G)+ 1 which has no cliquecutsets, no comparable pairs, and no universal cliques; 2) if G is a connectedimperfect $(P_7, C_5$, diamond)-free graph with \delta(G) \geq max{3,\omega(G)} and without comparable pairs, then G is isomorphic to a graph of awell defined 12 graph families; and 3) each connected imperfect $(P_7, C_5$,paw)-free graph is a blowup of $C_7$. As consequences, we show that \chi(G)\leq \omega(G)+1 if G is (P7, C5, kite, paraglider)-free, and \chi(G) \leqmax{3, \omega(G)} if G is $(P_7, C_5$, H)-free with H being a diamond or a paw.We also show that \chi(G) \leq

Structure and coloring of ($P_7$, $C_5$, diamond)-free graphs