Gallai asked in 1984 if any k-critical graph on n vertices contains at least n distinct (k − 1)-critical subgraphs. The answer is trivial for k ≤ 3. Improving a result of Stiebitz [10], Abbott and Zhou [1] proved in 1995 that for all k ≥ 4, any k-critical graph contains Ω(n1/(k − 1)) distinct (k − 1)-critical subgraphs. Since then no progress had been made until very recently, when Hare [4] resolved the case k = 4 by showing that any 4-critical graph on n vertices contains at least (8n − 29)/3 odd cycles.In this paper, we mainly focus on 4-critical graphs and develop some novel tools for counting cycles of specified parity. Our main result shows that any 4-critical graph on n vertices contains Ω(n2) odd cycles, which is tight up to a constant factor by infinitely many graphs. As a crucial step, we prove the same bound for 3-connected non-bipartite graphs, which may be of independent interest. Using the tools, we also give a short solution to Gallai’s problem when k = 4. Moreover, we improve the longstanding lower bound of Abbott and Zhou to Ω(n1/(k − 2)) for the general case k ≥ 5. We will also discuss some related problems on k-critical graphs in the final section.

For a vector space \({\mathbb{F}^n}\) over a field \(\mathbb{F}\), an (η, β)-dimension expander of degree d is a collection of d linear maps \({\Gamma _j}:{\mathbb{F}^n} \rightarrow {\mathbb{F}^n}\) such that for every subspace U of \({\mathbb{F}^n}\) of dimension at most ηn, the image of U under all the maps, ∑ dj=1 Γj(U), has dimension at least α dim(U). Over a finite field, a random collection of d = O(1) maps Γj offers excellent “lossless” expansion whp: β≈d for η ≥ Ω(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor β = 1+ ε with constant degree is a non-trivial goal.We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list decoding in the rank metric. Our approach yields the following: Lossless expansion over large fields; more precisely β ≥ (1 − ε)d and \(\eta \ge {{1 - \varepsilon} \over d}\) with d = Oε(1), when \(\left| \mathbb{F} \right| \ge \Omega \left(n \right)\). Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely β ≥ Ω(δd) and η ≥ Ω(1/(δd)) with d = Oδ(1), when \(\left| \mathbb{F} \right| \ge {n^\delta}\). Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Ω(1), 1 + Ω(1))-dimension expanders of constant degree over all fields. An approach based on “rank condensing via subspace designs” led to dimension expanders with \(\beta \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} \sqrt d \) over large finite fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.

The Hall ratio of a graph G is the maximum of |V(H)|/α(H) over all subgraphs H of G. It is easy to see that the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once). For every c > 0, every graph of sufficiently large average degree contains as a subgraph the 1-subdivision of a graph of fractional chromatic number at least c. For every d > 0, there exists a graph G of average degree at least d such that every graph whose 1-subdivision appears as a subgraph of G has Hall ratio at most 18. We also discuss the consequences of these results in the context of graph classes with bounded expansion.

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n?The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multi-normal of order \(1/\sqrt n \), while the orthogonal component is smaller.Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to Sk-representations.Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

Let X = (VX, EX) be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and EX is equipped with an involution which inverts the orientation. Each oriented edge is labelled by an element of a finite alphabet Σ. The labelling is assumed to be deterministic: edges with the same initial (resp. terminal) vertex have distinct labels. Furthermore, it is assumed that the group of label-preserving automorphisms of X acts quasi-transitively. For any vertex o of X, consider the language of all words over Σ which can be read along self-avoiding walks starting at o. We characterize under which conditions on the graph structure this language is regular or context-free. This is the case if and only if the graph has more than one end, and the size of all ends is 1, or at most 2, respectively.

We say that two graphs on the same vertex set are G-creating if their union (the union of their edges) contains G as a subgraph. Let Hn(G) be the maximum number of pairwise G-creating Hamiltonian paths of Kn. Cohen, Fachini and Korner proved $${n^{\frac{1}{2}n - o\left( n \right)}} \le {H_n}\left( {{C_4}} \right) \le {n^{\frac{3}{4}n + o\left( n \right)}}.$$ In this paper we close the superexponential gap between their lower and upper bounds by proving $${n^{\frac{1}{2}n - \frac{1}{2}\frac{n}{{\log n}} - O\left( 1 \right)}} \le {H_n}\left( {{C_4}} \right) \le {n^{\frac{1}{2}n + o\left( {\frac{n}{{\log n}}} \right)}}.$$ We also improve the previously established upper bounds on {enH}n({enC}2k) for k >3, and we present a small improvement on the lower bound of Furedi, Kantor, Monti and Sinaimeri on the maximum number of so-called pairwise reversing permutations. One of our main tools is a theorem of Krivelevich, which roughly states that (certain kinds of) good expanders contain many Hamiltonian paths.

We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high order walks is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs. In this work we go beyond spectral gap, and relate the shrinkage of a k-cochain by the walk operator, to its structure under the assumption of local spectral expansion. A simplicial complex is called a one-sided local spectral expander, if its links have large spectral gaps and a two-sided local spectral expander if its links have large two-sided spectral gaps. We show two Decomposition Theorems (one per one-sided/two-sided local spectral assumption): For every k-cochain ϕ defined on an n-dimensional local spectral expander, there exists a decomposition of ϕ into “orthogonal” parts that are, roughly speaking, the “projections” on the j-dimensional cochains for 0 ≤ j ≤ k. The random walk shrinks each of these parts by a factor of $$\frac{k+1-j}{k+2}$$ plus an error term that depends on the spectral expansion. When assuming one-sided local spectral gap, our Decomposition Theorem yields an optimal mixing for the high order random walk operator. Namely, negative eigenvalues of the links do not matter! This improves over [5] that assumed two-sided spectral gap in the links to get optimal mixing. This improvement is crucial in a recent breakthrough [1] proving a conjecture of Mihail and Vazirani. Additionally, we get an optimal mixing for high order random walks on Ramanujan complexes (whose links are one-sided local spectral expanders). When assuming two-sided local spectral gap, our Decomposition Theorem allows us to describe the whole spectrum of the random walk operator (up to an error term that is determined by the spectral gap) and give an explicit orthogonal decomposition of the spaces of cochains that approximates the decomposition to eigenspaces of the random walk operator.

In this paper, we extend and refine previous Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$ vertices of the clique, and let $f(n,k,c)=e(W_{n,k,c})$. Improving a celebrated theorem of Erd\H{o}s and Gallai, Kopylov proved that for $c<n$, any 2-connected graph $G$ on $n$ vertices with circumference $c$ has at most $\max{f(n,2,c),f(n,\lfloor\frac{c}{2}\rfloor,c)}$ edges. Recently, F\"uredi et al. proved a stability version of Kopylov's theorem. Their main result states that if $G$ is a 2-connected graph on $n$ vertices with circumference $c$ such that $10\leq c<n$ and $e(G)>\max{f(n,3,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)}$, then either $G$ is a subgraph of $W_{n,2,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, or $c$ is odd and $G$ is a subgraph of a member of two well-characterized families which we define as $\mathcal{X}_{n,c}$ and $\mathcal{Y}_{n,c}$. We prove that if $G$ is a 2-connected graph on $n$ vertices with minimum degree at least $k$ and circumference $c$ such that $10\leq c<n$ and $e(G)>\max{f(n,k+1,c),f(n,\lfloor\frac{c}{2}\rfloor-1,c)}$, then one of the following holds: (i) $G$ is a subgraph of $W_{n,k,c}$ or $W_{n,\lfloor\frac{c}{2}\rfloor,c}$, (ii) $k=2$, $c$ is odd, and $G$ is a subgraph of a member of $\mathcal{X}_{n,c}\cup \mathcal{Y}_{n,c}$, or (iii) $k\geq 3$ and $G$ is a subgraph of the union of a clique $K_{c-k+1}$ and some cliques $K_{k+1}$'s, where any two cliques share the same two vertices. This provides a unified generalization of the above result of F\"uredi et al. as well as a recent result of Li et al. and independently, of F\"uredi et al. on non-Hamiltonian graphs. Moreover, we prove a stability result on a classical theorem of Bondy on the circumference.

We investigate infinite highly-arc-transitive digraphs with two additional properties, property Z and descendant-homogeneity. We show that if D is a highly-arc-transitive descendant-homogeneous digraph with property Z and F is the subdigraph spanned by the descendant sets of a line in D, then F is a locally finite 2-ended digraph with property Z. If, moreover, D has prime out-valency, then there is only one possibility for the subdigraph F. This knowledge is then used to classify the highly-arc-transitive descendant-homogeneous digraphs of prime out-valency with property Z.