On potentially $K_{r+1}-U$-graphical Sequences

Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_m$). We use the symbol $Z_4$ to denote $K_4-P_2\mathrm{.}$ A sequence $S$ is potentially $K_m-H$-graphical if it has a realization containing a $K_m-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$denotethesmallestdegreesumsuchthatevery$n$-termgraphicalsequence$S$with$\sigma(S)\geq \sigma(K_{m}-H, n)$ispotentially$K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U,n)$ for $n\ge 5r+18,r+1\ge k\ge 7,$ $j\ge 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3\bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} \bigcup P_{3})$butnotcontainsacycleon4verticesandnotcontains$Z_4$. (for example, $C_3\bigcup C_{i_1}\bigcup C_{i_2}\bigcup >\mathrm{.}\mathrm{.}\mathrm{.}\bigcup C_{i_p}$ $(i_j\mathrm{\ne }4,j=2,3,\mathrm{.}\mathrm{.}\mathrm{.},p,i_1\ge 5)$, $C_3\bigcup P_{i_1} \bigcup P_{i_2} \bigcup … \bigcup P_{i_p}$ $(i_1\ge 3)$, $C_3\bigcup P_{i_1} \bigcup C_{i_2} \bigcup >… \bigcup C_{i_p}$ $(i_j\mathrm{\ne }4,j=2,3,\mathrm{.}\mathrm{.}\mathrm{.},p,i_1\ge 3)$, etc)