KAIST Discrete Math Seminar

Say that a graph with maximum degree at most d is d-bounded. For d>k, we prove a sharp sparseness condition for decomposability into k forests and a d-bounded graph. Consequences ar e that every graph with fractional arboricity at most k+ d/(k+d+1) has such a decomposition, and (for k=1) every graph with maximum average degree less than 2+2d/(d+2) decomposes into a forest and a d-bounded graph. When d=k+1, and when k=1 and d≤6, the d-bounded graph in the decomposition can also be required to be a forest. When k=1 and d≤2, the d-bounded forest can also be required to have at most d edges in each component.
This is joint work with A.V. Kostochka, D.B. West, H. Wu, and X. Zhu.

A transversal matroid is a collection of objects that encodes maximal matchings in a bipartite graph. Generalized permutohedra is a class of polytopes obtained by deforming the permutohedron. We introduce a nice bijection that allows one to view transversal matroids as set of lattice points inside a generalized permutohedron. As a corollary, we solve a special case of the 30-year old conjecture by Stanley on matroids and pure O-sequences. The talk will be elementary and purely combinatorial.

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.
This is joint work with Boris Alexeev, and Alexandra Fradkin.