Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

Abstract

Abstract. We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

Iterated claws have real-rooted genus polynomials

“…Thus, even though Stahl's proposed approach to log-concavity via roots of genus polynomials is sometimes infeasible, [7] does support Stahl's expectation that chains of copies of a graph are a relatively accessible aspect of the general LCGD problem. Moreover, Wagner [14] has proved the real-rootedness of the genus polynomials for a number of graph families for which Stahl made specific conjectures of real-rootedness.…”

“…The need for more powerful techniques motivated the development of the linear combination techniques of [7]. Here, to prove the log-concavity of the genus polynomials for the sequence of iterated claws, we combine Newton's theorem that a real-rooted polynomial is log-concave (Theorem 4.1) with a focus on interlacing of roots of consecutive genus polynomials for the graphs in the sequence to prove their log-concavity.…”

“…We observe that, whereas all of Stahl's examples [12] of graphs with log-concave genus distributions are planar, the sequence of iterated claws has rising minimum genus. (Example 3.2 of [7] is another sequence of rising minimum genus. However, the graphs in that sequence have cutpoints, unlike the iterated claws.…”

Linear Time Canonicalization and Enumeration of Non-Isomorphic 1-Face Embeddings

“…Enumeration of embeddings, both with and without regard to isomorphism, is a classical topic which attracted a significant amount of attention, see for example [15,16,35,36]. However, the enumeration of embeddings is difficult and very little progress has been made in enumerating pairwise non-isomorphic embeddings.…”

“…maximum) genus of an orientable surface into which the graph can be cellularly embedded. The minimum and maximum genus and, more generally, the genus distribution -the number of embeddings of G into each surface into which it can be embedded, are arguably the most fundamental topological invariants of a graph and a large body of research is devoted to them, see for example [13,15,16,19,22,31,37].…”