Given a gene tree topology and a species tree topology, a coalescent history represents a possible mapping of the list of gene tree coalescences to associated branches of a species tree on which those coalescences take place. Enumerative properties of coalescent histories have been of interest in the analysis of relationships between gene trees and species trees. The simplest enumerative result identifies a bijection between coalescent histories for a matching caterpillar gene tree and species tree with monotonic paths that do not cross the diagonal of a square lattice, establishing that the associated number of coalescent histories for n-taxon matching caterpillar trees (n 2) is the Catalan number C n−1 = 1 n 2n−2 n−1 . Here, we show that a similar bijection applies for non-matching caterpillars, connecting coalescent histories for a non-matching caterpillar gene tree and species tree to a class of roadblocked monotonic paths. The result provides a simplified algorithm for enumerating coalescent histories in the non-matching caterpillar case. It enables a rapid proof of a known result that given a caterpillar species tree, no non-matching caterpillar gene tree has a number of coalescent histories exceeding that of the matching gene tree. Additional results on coalescent histories can be obtained by a bijection between permissible roadblocked monotonic paths and Dyck paths. We study the number of coalescent histories for non-matching caterpillar gene trees that differ from the species tree by nearest-neighbor-interchange and subtree-prune-and-regraft moves, characterizing the non-matching caterpillar with the largest number of coalescent histories. We discuss the implications of the results for the study of the combinatorics of gene trees and species trees.

We study the following question: How few edges can we delete from any H $H$‐free graph on n $n$ vertices to make the resulting graph k $k$‐colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H $H$ any fixed odd cycle, we determine the answer up to a constant factor when n $n$ is sufficiently large. We also prove an upper bound when H $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.

We study the following question: how few edges can we delete from any H-free graph on n vertices in order to make the resulting graph k-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H any fixed odd cycle, we determine the answer up to a constant factor when n is sufficiently large. We also prove an upper bound when H is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.

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Given a graph sequence and a simple connected subgraph , we denote by the number of monochromatic copies of in a uniformly random vertex coloring of with colors. We prove a central limit theorem for (we denote the appropriately centered and rescaled statistic as ) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of which we call good joins. Good joins are closely related to the fourth moment of , which allows us to show a fourth moment phenomenon for the central limit theorem. For , we show that converges in distribution to whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when .