Dendrites or lambda-dendroids as generalized inverse limits

This research centers on the study of generalized inverse limits. We show that all members of an infinite family of inverse limit spaces are homeomorphics to one particularly complicated inverse limit space known as The Monster . Further, properties of factor spaces and graphs of bonding functions which are preserved in generalized inverse limit spaces with upper semi-continuous bonding functions with appropriate restrictions are investigated. Some of the properties are locally connectedness, hereditary decomposability, hereditary indecomposability, hereditary unicoherence, arc-likeness, and tree-likeness. The theorems are illustrated by several examples --Abstract, page iv

On matchings and factors of graphs /

In Section 1, we recall the historical sketch of matching and factor theory of graphs, and also introduce some necessary definitions and notation. In Section 2, we present a sufficient condition for the existence of a (g, f)-factor in graphs with the odd-cycle property, which is simpler than that of Lovasz\u27s (g, f)-Factor Theorem. From this, we derive some further results, and we show that (a) every r-regular graph G with the odd-cycle property has a k-factor, where 0 ≤ k ≤ r and k|V(G)| ≡ 0 (mod 2), (b) every graph G with the strong odd-cycle property with k|V(G)|≡ 0 (mod 2) is k-factorable if and only if G is a km-regular graph for some m ≥ 1, and (c) every regular graph of even order with the strong odd-cycle property is of the second class (i.e. the edge chromatic number is Δ). Chvátal [26] presented the following two conjectures that (1) a graph G has a 2-factor if tough(G) ≥ 3/2, and (2) a graph G has a k-factor if k|V(G)| ≡ 0 (mod 2) and tough(G) ≥ k. Enomoto et.al. [32] proved the second conjecture. They also proved the sharpness of the bound on tough(G) that guarantees the existence of a k-factor. This implies that the first conjecture is false. In Section 3, we show that the result of the second conjecture can be improved in some sense, and the first conjecture is also true if the graph considered has the odd-cycle property. Anderson [3] stated that a graph G of even order has a 1-factor if bind(G) ≥ 4/3, and Katerinis and Woodall [48] proved that a graph G of order n has a k-factor if bind(G) ˃ (2k -I)(n - 1)/(k(n - 2) + 3), where k ≥ 2, n ≥ 4k - 6 and kn ≡ 0 (mod 2). In Section 4, we shall present some similar conditions for the existence of [a, b]-factors. In Section 5, we study the existence of [a, b]-parity-factors in a graph, among which we extend some known theorems from 1-factors to {1, 3, ... , 2n - 1}-factors, or from k-factors to [a, b]-parity-factors. Also, extending Petersen\u27s 2-Factorization Theorem, we proved that a graph is [2a, 2b]-even-factorable if and only if it is a [2na, 2nb]-even-graph for some n ≥ 1. Plummer showed that (a) (in [58]) every graph G of even order is k-extendable if tough(G) ˃ k, and (b) (in [59]) every (2k+1)-connected graph G is k-extendable if G is K1,3-free, respectively. In Section 6, we give a counterpart of the former in terms of binding number, and extend the latter from K1,3-free graphs to K1,n-free graphs. Furthermore, we present a result toward the problem, posed by Saito [61] and Plummer [60], of characterizing the graphs that are maximal k-extendable

Central limit theorems for patterns in multiset permutations and set partitions

We use the recently developed method of weighted dependency graphs to prove central limit theorems for the number of occurrences of any fixed pattern in multiset permutations and in set partitions. This generalizes results for patterns of size 2 in both settings, obtained by Canfield, Janson and Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.Comment: version 2 (52 pages) implements referee's suggestions and uses journal layou

Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Friedland's Lower Matching Conjecture asserts that if $G$ is a $d$--regular bipartite graph on $v(G)=2n$ vertices, and $m_k(G)$ denotes the number of matchings of size $k$, then $$m_k(G)\geq {n \choose k}^2\left(\frac{d-p}{d}\right)^{n(d-p)}(dp)^{np},$$ where $p=\frac{k}{n}$. When $p=1$, this conjecture reduces to a theorem of Schrijver which says that a $d$--regular bipartite graph on $v(G)=2n$ vertices has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$ perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that $$\frac{\ln m_k(G)}{v(G)}\geq \frac{1}{2}\left(p\ln \left(\frac{d}{p}\right)+(d-p)\ln \left(1-\frac{p}{d}\right)-2(1-p)\ln (1-p)\right)+o_{v(G)}(1).$$ In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra $c_p\sqrt{n}$ factor compared to the conjecture if $p$ is separated away from $0$ and $1$, and is tight up to a constant factor if $p$ is separated away from $1$. We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to $(a,b)$--biregular bipartite graphs