MATCHINGS IN REGULAR GRAPHS: MINIMIZING THE PARTITION FUNCTION

For a graph G on v(G) vertices let m(k)(G) denote the number of matchings of size k, and consider the partition function M-G(lambda) = Sigma(n)(k=0)m(k)(G)lambda(k). In this paper we show that if G is a d-regular graph and 0 1/v(Kd+1) ln MKd+1(lambda).The same inequality holds true if d = 3 and lambda < 0.3575. More precise conjectures are also given

Matchings in Benjamini–Schramm convergent graph sequences

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini– Schramm convergent sparse graph sequence. We prove that the normalized loga-rithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d–regular bipartite graphs for any fixed d ≥ 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of d–regular bipartite graphs converging to the d– regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d–regular bipartite graphs. We show that the limit equals to the exponent in Schrijver’s lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen–Onak (also Elek– Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs

Short Proof of a Theorem of Brylawski on the Coefficients of the Tutte Polynomial

In this short note we show that a system $M=(E,r)$ with a ground set $E$ of size $m$ and (rank) function $r: 2^E\to \mathbb{Z}_{\geq 0}$ satisfying $r(S)\leq \min(r(E),|S|)$ for every set $S\subseteq E$, the Tutte polynomial $$T_M(x,y):=\sum_{S\subseteq E}(x-1)^{r(E)-r(S)}(y-1)^{|S|-r(S)},$$ written as $T_M(x,y)=\sum_{i,j}t_{ij}x^iy^j$, satisfies that for any integer $h \geq 0$, we have $$\sum_{i=0}^h\sum_{j=0}^{h-i}\binom{h-i}{j}(-1)^jt_{ij}=(-1)^{m-r}\binom{h-r}{h-m},$$ where $r=r(E)$, and we use the convention that when $h<m$, the binomial coefficient $\binom{h-r}{h-m}$ is interpreted as $0$. This generalizes a theorem of Brylawski on matroid rank functions and $h<m$, and a theorem of Gordon for $h\leq m$ with the same assumptions on the rank function. The proof presented here is significantly shorter than the previous ones. We only use the fact that the Tutte polynomial $T_M(x,y)$ simplifies to $(x-1)^{r(E)}y^{|E|}$ along the hyperbola $(x-1)(y-1)=1$.Comment: 4 page

Analitikus és kombinatórikus számelmélet = Analytical and combinatorial number theory

A kutatás négy és fél éve alatt a résztvevők 87 tudományos dolgozatot írtak. Valamennyi cikk angol nyelvű. 81 dolgozat erős folyóiratban (eltekintve 3 Eötvös Annales cikktől), a maradék 6 cikk speciális alkalomra kiadott könyvben (cikkgyűjteményben) jelent meg. A 87 cikk közül 40 (46%) olyan volt, melynek társszerzői közt legalább egy külföldi volt; a kapott OTKA támogatás nagy mértékben hozzájárult e cikkek megszületéséhez. A 87 dolgozat közül 57 (65%) jelent meg külföldön. A legfontosabb cikkek témája prímszámelmélet (Pintz János és társszerzői), valamilyen értelemben rendezett struktúrák (bináris sorozatok, bináris ""gyökeres"" fák, rendezett halmazok részhalmazai, bináris vektorok) pszeudovéletlensége (Gyarmati Katalin, Mérai László, Sárközy András és társszerzőik), egész számok sorozatainak additív tulajdonságai (Gyarmati Katalin, Hegyvári Norbert, Károlyi Gyula, Ruzsa Z. Imre, Sárközy András és társszerzőik), véges testek számelmélete (Gyarmati Katalin, Sárközy András és társszerzőik), valamint moduláris formák (Biró András) volt. | During the four and half years of the project 87 research papers have been written by the participants. All these papers are of English language. 81 papers have been published in strong journals (apart from 3 papers published in the Eötvös Annales) while the remaining 6 papers have appeared in books published on some special occasion. 40 papers (46%) out of the 87 papers have at least one non-Hungarian coauthor; this fund largely contributed to the preparation of these papers. 57 (65%) of the 87 papers have appeared abroad. The most important papers have been written on prime number theory (by J. Pintz and his coauthors), pseudorandomness of structures (binary sequences, $k$-symbol sequences, lattices, binary rooted trees, subsets of ordered sets, binary vectors with some sort ordering (by K. Gyarmati, L. Mérai, A. Sárközy and coauthors), additive properties of sequences of integers (by K. Gyarmati, N. Hegyvári, Gy. Károlyi, I. Z. Ruzsa, A. Sárközy and coauthors), arithmetics of finite fields (K. Gyarmati, A. Sárközy and coauthors) and modular forms (A. Biró)

Upper bound for the number of spanning forests of regular graphs

We show that if G is a d–regular graph on n vertices, then the number of spanning forests F (G) satisfies F (G) ≤ dn. The previous best bound due to Kahale and Schulman gave (d+1/2+O(1/d))n. We also have the more precise conjecture that F (G)1/n ≤ (d − 1)d−1 (d2 − 2d − 1)d/2−1 . If this conjecture is true, then the expression on the right hand side is the best possible