the condition for a sequence to be potentially $a_{l, m}$- graphic
نویسندگان
چکیده
the set of all non-increasing non-negative integer sequences $pi=(d_1, d_2,ldots,d_n)$ is denoted by $ns_n$. a sequence $piin ns_{n}$ is said to be graphic if it is the degree sequence of a simple graph $g$ on $n$ vertices, and such a graph $g$ is called a realization of $pi$. the set of all graphic sequences in $ns_{n}$ is denoted by $gs_{n}$. the complete product split graph on $l + m$ vertices is denoted by $overline{s}_{l, m}=k_{l} vee overline{k}_{m}$, where $k_{l}$ and $k_{m}$ are complete graphs respectively on $l = sumlimits_{i = 1}^{p}r_{i}$ and $m = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. another split graph is denoted by $s_{l, m} = overline{s}_{r_{1}, s_{1}} veeoverline{s}_{r_{2}, s_{2}} vee cdots vee overline{s}_{r_{p}, s_{p}}= (k_{r_{1}} vee overline{k}_{s_{1}})vee (k_{r_{2}} vee overline{k}_{s_{2}})vee cdots vee (k_{r_{p}} vee overline{k}_{s_{p}})$. a sequence $pi=(d_{1}, d_{2},ldots,d_{n})$ is said to be potentially $s_{l, m}$-graphic (respectively $overline{s}_{l, m}$)-graphic if there is a realization $g$ of $pi$ containing $s_{l, m}$ (respectively $overline{s}_{l, m}$) as a subgraph. if $pi$ has a realization $g$ containing $s_{l, m}$ on those vertices having degrees $d_{1}, d_{2},ldots,d_{l+m}$, then $pi$ is potentially $a_{l, m}$-graphic. a non-increasing sequence of non-negative integers $pi = (d_{1}, d_{2},ldots,d_{n})$ is potentially $a_{l, m}$-graphic if and only if it is potentially $s_{l, m}$-graphic. in this paper, we obtain the sufficient condition for a graphic sequence to be potentially $a_{l, m}$-graphic and this result is a generalization of that given by j. h. yin on split graphs.
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عنوان ژورنال:
transactions on combinatoricsجلد ۶، شماره ۱، صفحات ۲۱-۲۷
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