A. A. Mironov Russ. Acad. Sci., Dokl. Math. , vol.48 No.3 , 01.1994 , p. 566-569, language: английский ISSN 1064-5624  Аннотация The degree of a vertex of a generalized graph (multigraph or weighted graph with or without loops) is the sum of the multiplicities or weights of edges (loops) incident to this vertex. For $n\ge 2$ we denote $$\overline R\sp n= \{A= \{a\sb 1,\dots, a\sb n\}: a\sb i\ge a\sb{i+ 1},\ 1\le i\le n- 1,\a\sb n\ge 0\}$$ and $$\overline Z\sp n= \{A\in \overline R\sp n: a\sb i\in Z,\ 1\le i\le n\}.$$ A set $A$ of numbers in $\overline Z\sp n$ or $\overline R\sp n$ is called realizable ($c$-realizable) in a generalized graph whose multiplicities or weights of edges (loops) do not exceed $c\ge 0$, if $A$ is the set of degrees of vertices of some $G= G(A)$ in $\Gamma(c)\cup \Gamma\sb L(c)\cup \Gamma\sb Z(c)\cup \Gamma\sb{ZL}(c)$, where $\Gamma(c)$ is the class of weighted graphs without loops, the index $L$ indicates the class with loops, and the index $Z$ the class of multigraphs. $G(A)$ is called a realization ($c$-realization) of $A$. We define $$\overline R\sp n\sb r(c)= \{A\in \overline R\sp n: \exists G(A)\in \Gamma(c)\}\text{and}\overline R\sp n\sb{rL}(c)= \{A\in \overline R\sp n: \exists G(A)\in \Gamma\sb L(c)\},$$ and if $c\in Z$, then $$\overline Z\sp n\sb r(c)= \{A\in \overline Z\sp n: \exists G(A)\in \Gamma\sb Z(c)\}\text{and} \overline Z\sp n\sb{rL}(c)= \{A\in \overline Z\sp n: \exists G(A)\in \Gamma\sb{ZL}(c)\}.$$ It is obvious that if $A$ is $c$-realizable, then $A$ is $d$-realizable for any $d> c$. In the aforementioned paper we constructed systems of functions depending on $A= \{a\sb 1,\dots, a\sb n\}$, $c> 0$, and $k\in Z$, $1\le k\le n- 1$, whose nonnegativity is a criterion for the $c$-realizability of $A$, and we defined the smallest values of $c$ for which $A\in \overline R\sp n\sb r(c)$, $\overline R\sp n\sb{rL}(c)$, $\overline Z\sp n\sb r(c)$, $\overline Z\sp n\sb{rL}(c)$.\par In the present note we give unimprovable sufficient conditions for $c$- realizability based on the fulfillment of one inequality depending on the three parameters $n$, $a\sb n$, and $c$, and we define necessary and sufficient conditions for realizability of $A\in \overline R\sp n$ and $B\in \overline R\sp m$ in a bipartite generalized graph.  Ключевые слова degree of a vertex; generalized graph; weights; realizable; realization; realizability; bipartite generalized graph