Classification of degree (bi-)sequences of bipartite graphs? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T16:32:59Z http://mathoverflow.net/feeds/question/48702 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48702/classification-of-degree-bi-sequences-of-bipartite-graphs Classification of degree (bi-)sequences of bipartite graphs? JBL 2010-12-09T03:08:03Z 2010-12-09T19:27:08Z <p>It is known that the sequence $d_1 \geq d_2 \geq \ldots \geq d_n$ of nonnegative integers is the degree sequence of a graph if and only if the sum of the $d_i$ is even and we have $\sum_{i = 1}^k d_i \leq k(k - 1) + \sum_{i = k + 1}^n \min(d_i, k)$ for all $k \in \{1, \ldots, n\}$. (This is the Erdős–Gallai theorem.) There is also a simple algorithm (the Havel-Hakimi algorithm) for producing a graph with a given valid degree sequence.</p> <blockquote> <p>Is there a simple characterization of pairs $\{(a_1 \geq \ldots \geq a_m), (b_1 \geq \ldots \geq b_n)\}$ of sequences of nonnegative integers such that there exists a bipartite graph with the property that the degrees of the vertices in one part are given by the $a_i$ and the degrees of the vertices in the other part is given by the $b_j$?</p> </blockquote> <p>An obvious necessary condition is that $\sum a_i = \sum b_j$, but this is also clearly not sufficient. One can also ask for an algorithm analogous to Havel-Hakimi.</p> http://mathoverflow.net/questions/48702/classification-of-degree-bi-sequences-of-bipartite-graphs/48703#48703 Answer by Igor Rivin for Classification of degree (bi-)sequences of bipartite graphs? Igor Rivin 2010-12-09T03:22:06Z 2010-12-09T03:22:06Z <p>This is only a semi-answer: The existence of such a graph is equivalent to finding the maximal flow in a network where the source is connected to all the "red" vertices by edges of capacities $a_1, \dots, a_n$, the sink is connected to all the "blue" vertices by edges of capacity $b_1, \dots, b_n$ and every blue edge is connected to every red edge by an edge of capacity $1.$ If the maximal flow equals $\sum a_i = \sum b_j,$ then the degree sequence exists. Finding such a flow is quite fast (see, eg, Ahuja and Orlin's book on network flow).</p> http://mathoverflow.net/questions/48702/classification-of-degree-bi-sequences-of-bipartite-graphs/48713#48713 Answer by Gjergji Zaimi for Classification of degree (bi-)sequences of bipartite graphs? Gjergji Zaimi 2010-12-09T05:06:36Z 2010-12-09T05:06:36Z <p>So you are looking for the <a href="http://mathworld.wolfram.com/Gale-RyserTheorem.html" rel="nofollow">Gale-Ryser theorem</a>. There is a version of Havel-Hakimi for bipartite graphs as well. It says that the pair $(P,Q)$ is bigraphic if and only if the pair $(P',Q')$ is bigraphic, where $(P', Q')$ is obtained from $(P, Q)$ by deleting the largest element $p_1$ of $P$ and subtracting one from each of the $p_1$ largest elements of $Q$.</p>