Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced

The chromatic index of strongly regular graphs

We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree $k \leq 18$ and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements. Moreover, using a recent result of Ferber and Jain it is shown that an SRG of even order $n$, which is not the block graph of a Steiner 2-design or its complement, has chromatic index $k$, when $n$ is big enough. Except for the Petersen graph, all investigated connected SRGs of even order have chromatic index equal to their degree, i.e., they are class 1, and we conjecture that this is the case for all connected SRGs of even order.Comment: 10 page