In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves and show that there is exactly one Wilf class of k-leaf tree patterns for any positive integer k. In addition, we enumerate binary trees that simultaneously avoid more than one tree pattern. Finally, we explore connections between pattern-avoiding trees and pattern-avoiding permutations.
We conjecture that all connected graphs of order n have von Neumann entropy at least as great as the star K 1,n−1 and prove this for almost all graphs of order n. We show that connected graphs of order n have Rényi 2-entropy at least as great as K 1,n−1 and for α > 1, K n maximizes Rényi α-entropy over graphs of order n. We show that adding an edge to a graph can lower its von Neumann entropy.
Abstract. The Shi arrangement is the set of all hyperplanes in R n of the form xj − x k = 0 or 1 for 1 ≤ j < k ≤ n. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is (n + 1) n−1 . An unrelated combinatorial concept is that of a parking function, i.e., a sequence (x1, x2, ..., xn) of positive integers that, when rearranged from smallest to largest, satisfies x k ≤ k. (There is an illustrative reason for the term parking function.) It turns out that the number of parking functions of length n also equals (n + 1) n−1 , a result due to Konheim and Weiss from 1966. A natural problem consists of finding a bijection between the n-dimensional Shi arragnement and the parking functions of length n. Pak and Stanley (1996) and Athanasiadis and Linusson (1999) gave such (quite different) bijections. We will shed new light on the former bijection by taking a scenic route through certain mixed graphs.
All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ℓ ∈ {3, 4, 5, 6, 7}, a planar graph is 4-choosable if it is ℓ-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ ∈ {5, 6, 7} that a planar graph is (4, 2)-choosable if it does not contain chorded ℓ-cycles.
Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph G begin in one of two states, “dormant” or “active.” Given a fixed positive integer r, a dormant vertex becomes active if at any stage it has at least r active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices A, we say that G r‐percolates (from A) if every vertex in G becomes active after some number of steps. Let m(G,r) denote the minimum size of a set A such that G r‐percolates from A. Bootstrap percolation has been studied in a number of settings and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree‐based density conditions that ensure m(G,2)=2. In particular, we give an Ore‐type degree sum result that states that if a graph G satisfies σ2(G)≥n−2, then either m(G,2)=2 or G is in one of a small number of classes of exceptional graphs. (Here, σ2(G) is the minimum sum of degrees of two nonadjacent vertices in G.) We also give a Chvátal‐type degree condition: If G is a graph with degree sequence d1≤d2≤⋯≤dn such that di≥i+1 or dn−i≥n−i−1 for all 1≤i
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