The degree sequences of a graph with restrictions

Two finite sequences s1 and s2 of nonnegative integers are called bigraphical if there exists a bipartite graph G with partite sets V1 and V2 such that s1 and s2 are the degrees in G of the vertices in V1 and V2, respectively. In this paper, we introduce the concept of 1-graphical sequences and present a necessary and sufficient condition for a sequence to be 1-graphical in terms of bigraphical sequences.


Introduction
We generally follow the notation and terminology pertaining to graphs of [1]. If F is a nonempty subset of the edge set E (G) of a graph G, then the subgraph F induced by F is www.ijc.or.id The degree sequences of ... | R. Ichishima, F. A. Muntaner-Batle, M. Rius-Font, and Y. Takahashi the graph whose vertex set consists of those vertices of G incident with at least one edge of F and whose edge set is F . The degree of a vertex v in a graph G is the number of edges of G incident with v, which is denoted by deg v. A vertex is called even or odd according to whether its degree is even or odd.
A sequence d 1 , d 2 , . . . , d n of nonnegative integers is called a degree sequence of a graph G if the vertices of G can be labeled v 1 , v 2 , . . . , v n so that deg v i = d i for all i. We adopt the convention that the vertices have been labeled so that d 1 ≥ d 2 ≥ · · · ≥ d n . We call a sequence of nonnegative integers graphical if it is the degree sequence of some graph. A necessary and sufficient condition for a sequence to be graphical was found by Havel [4] and later rediscovered by Hakimi [3].
According to the definition of a simple graph, two distinct vertices are joined by at most one edge. If we allow more than one edge (but a finite number) to join pairs of vertices, the resulting structure is called a multigraph. If two or more edges join the same two vertices in a multigraph, then these edges are referred to as multiple edges. Hakimi [3] extended the preceding result to multigraphs.
Then there exists a multigraph with degree sequence s : Two finite sequences s 1 and s 2 of nonnegative integers are called bigraphical if there exists a bipartite graph G with partite sets V 1 and V 2 such that s 1 and s 2 are the degrees in G of the vertices in V 1 and V 2 , respectively. The following result is an analog of Theorem 1.1 for graphs (see [1, p. 16]). Theorem 1.3. The sequences s 1 : a 1 , a 2 , . . . , a r and A loop is an edge that joins a vertex to itself and contributes to the degree of a vertex twice. A graph G is called a 1-graph if it has at most one loop attached at each vertex and at most two multiple edges joining each pair of vertices. A sequence s is called 1-graphical if there exists a 1-graph that realizes s.
For the sake of notational convenience, we will denote the interval of integers x such that a ≤ x ≤ b by simply writing [a, b].
In this paper, we present a necessary and sufficient condition for a sequence to be 1-graphical in terms of bigraphical sequences. To this end, we use the following theorem, due to Veblen [7], which characterizes eulerian graphs in terms of their cycle structures. To conclude this introduction, it is worth to mention that López and Muntaner-Batle [5] completely characterized the degree sequences of graphs with at most one loop attached at each vertex and no multiple edges. Hence, the work conducted in this paper would be a natural continuation of their work.

Characterization of 1-graphical sequences
We are now ready to state and prove the following theorem. This result has the following consequences.