For a digraph G, let f (G) be the maximum chromatic number of an acyclic subgraph of G. For an n-vertex digraph G it is proved that f (G) > n(5/9-o(1))s(-14/9) where s is the bipartite independence number of G, i.e., the largest s for which there are two disjoint s-sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case s = 0 (i.e., tournaments and semicomplete digraphs). Consequently, if s = n(o(1)), then f (G) >= n(5/9-o(1)) which polynomially improves the folklore bound f (G) >= n(1/2-o(1)). As a corollary, with high probability, all orientations of the random n-vertex graph with edge probability p = n(-o(1)) (in particular, constant p, hence almost all n-vertex graphs) satisfy f (G) >= n(5/9-o(1)). Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

It follows from a classical result of Jordan that every tree with maximum degree at most r containing a vertex set labeled by [n], has a single-edge cut which separates two subsets A, B subset of [n] for which min{|A|, |B|} >= (n-1)/r. Motivated by the tree dissimilarity problem in phylogenetics, we consider the case of separating vertex sets of several trees: Given k trees with maximum degree at most r, containing a common vertex set labeled by [n], we ask for a single-edge cut in each tree which maximizes min{|A|, |B|} where A, B subset of [n] are separated by the corresponding cut at each tree. Denoting this maximum by f (r, k, n) and considering the limit f (r, k) = limn ->infinity f (r, k, n)/n (which is shown to always exist) we determine that f(r, 2) = 2r1 and determine that f (3, 3) = 227, which is already quite intricate. The case r = 3 is especially interesting in phylogenetics and our result implies that any two (three) binary phylogenetic trees over n taxa have a split at each tree which separates two taxa sets of order at least n/6 (resp. 2n/27), and these bounds are asymptotically tight. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

A king in an n-vertex tournament graph G is a vertex that can reach any other vertex v with a path of length at most two. A kingdom is a data structure that given any vertex v returns such a path from a king to v in O(1) time. In this paper, we show how to maintain a kingdom while the tournament graph G undergoes updates. We consider both edge updates that flip the direction of an edge, and vertex updates that insert/delete vertices by activating/deactivating rows and columns of the graph’s adjacency matrix. For a single edge-flip, we show that after O(n^3/2) preprocessing time, we can maintain a kingdom in O(1) time following the edge-flip. With Õ(n^2) preprocessing time, we can support any constant number of edge-flips, vertex insertions, and vertex deletions in O(log n) time per operation. For an arbitrary number of edge-flips, we present a randomized algorithm that maintains a kingdom in O(log n) expected time following every edge-flip, and another algorithm that supports vertex insertions in O(√(n)) amortized time per insertion.

The Erdős–Stone Theorem asserts that if a graph has edge density 1-1/r+δ then it contains a complete (r+1)-partite graph with b vertices in each part, where b=b_n(r,δ) ≫ 1. The celebrated Chvátal–Szemerédi theorem determined the exact order of b_n(r,δ) for every δ< 1/r^3. Their bound, however, is not tight when δ=1/r-ε, that is, when the graph has edge density 1-ε for small ε. Our main result in this paper determines the correct order in this remaining regime, thereby enabling us to give a tight bound for the Erdős–Stone problem for all edge densities. More precisely, we prove that for every integer r≥ 2 and 0< δ< 1/r we have b_n(r,δ)=Θ(log n/(1/r-δ)rlog(1/δ)) . The lower bound is obtained using a Kövari-Sós-Turán-type argument combined with a variant of Nikiforov's method of constructing large blow-ups, while the upper bound is proved using a correlated random graph construction, related to tensor powers.

We prove that the inducibility of P_4 in ordered monotone balanced bipartite graphs is 2/e^2, establishing the smallest known graph with transcendental Turán-type density. Moreover, the limit object is a binary graphon, so it generates a deterministic model. This is a special case of a more general framework addressed here – the asymptotic maximum density of a constant matrix over an arbitrary symbol set, in a large, possibly monotone, matrix. We solve all 2 × 2 monotone cases (one of which corresponds to the aforementioned P_4) and all but one of the 2 × 2 unrestricted cases. While (h!/h^h)^2 is a lower bound for the asymptotic maximum density of an h × h matrix, we explicitly construct, for all h ≥ 1, an h × h minimizer, i.e., a matrix for which this bound is attained. We also sketch how known results on the inducibility of graphs can be modified to show that, as h grows, almost all h × h 0/1 matrices are minimizers.

For graphs F and H, let i(F) denote the inducibility of F and let iH(F) denote the inducibility of F over H-free graphs. We prove that for almost all graphs F on a given number of vertices, iKk(F) attains infinitely many values as k varies. For complete partite graphs F (and, more generally, for symmetrizable families of graphs F), we prove that iH(F)=iKk(F) where k=χ(H), and is attained by a complete ℓ-partite graphon WF,k, where ℓ<k.We determine the part sizes of WF,k for all k, whence determine i(F), whenever F is the Turán graph on s vertices and r parts, for all s≤3r+1, which was recently proved by Liu, Mubayi, and Reiher for s=r+1. As a corollary, this determines the inducibility of all Turán graphs on at most 14 vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree 1, of any size. Similarly, this determines the inducibility of all triangle factors, of any size.For complete partite graphs F with at most one singleton part, we prove that iKk(F) only attains finitely many values as k varies; in particular, there exists t=t(F) such that i(F) is attained by some complete t-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts.Finally, for every r, we give a nontrivial sufficient condition for a complete r-partite graph F to have the property that i(F) is attained by a complete partite graphon all whose part sizes are distinct.

A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of tournaments, it is well known that every tournament contains a transitive subgraph of size $\log n$, and that this is best possible up to a constant factor. A restricted family of tournaments that has been extensively studied is the family of $k$-majority tournaments. They are obtained by taking $2k-1$ linear orders of a set $X$, and defining a tournament on $X$ which has an edge from $u$ to $v$ if $u$ precedes $v$ in at least $k$ of these orders. Milans, Schreiber, and West proved that such tournaments indeed have significantly larger transitive tournaments. More precisely, they proved that every $k$-majority tournament contains a transitive tournament of size $n^{2^{-Θ(k)}}$. Our main goal in this paper is to give an exponential improvement in the dependence of the exponent on $k$ by showing that every $k$-majority tournament contains a transitive set of size $n^{Ω(1/k)}$. Finally, we highlight several open problems and conjectural directions related to random $k$-majority tournaments.

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, it is proved that every tournament on four vertices (there are four) can be detected in $O(n^2)$ time and counted in $O(n^\omega)$ time where $\omega < 2.373$ is the matrix multiplication exponent. It is also proved that any tournament on five vertices (there are $12$) can be counted in $O(n^{\omega+1})$ time. As for lower-bounds, we prove that for almost all $k$-vertex tournaments, the complexity of the detection problem is not easier than the complexity of the corresponding well-studied counting problem for {\em undirected cliques} of order $k-O(\log k)$.

We consider the asymptotic minimum density $f(s,k)$ of monotone $k$-subwords of words over a totally ordered alphabet of size $s$. The unrestricted alphabet case, $f(\infty,k)$, is well-studied, known for $f(\infty,3)$ and $f(\infty,4)$, and, in particular, conjectured to be rational for all $k$. Here we determine $f(2,k)$ for all $k$ and determine $f(3,3)$, which is already irrational. We describe an explicit construction for all $s$ which is conjectured to yield $f(s,3)$. Using our construction and flag algebra, we determine $f(4,3),f(5,3),f(6,3)$ up to $10^{-3}$ yet argue that flag algebra, regardless of computational power, cannot determine $f(5,3)$ precisely. Finally, we prove that for every fixed $k \ge 3$, the gap between $f(s,k)$ and $f(\infty,k)$ is $\Theta(\frac{1}{s})$.

An inversion of a tournament T is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let inv(k)(T) be the minimum length of a sequence of inversions using sets of size at most k that result in the transitive tournament. Let inv(k)(n) be the maximum of inv(k)(T) taken over n-vertex tournaments. It is well known that inv(2)(n)=(1+o(1))n(2)/4 and it was recently proved by Alon et al. that & colone;inv(n)& colone;inv(n)(n)=n(1+o(1)). In these two extreme cases (k=2 and n), random tournaments are extremal objects. It is proved that inv(k)(n) is not attained by random tournaments when k >= k(0) and conjectured that inv(3)(n) is (only) attained by (quasi)random tournaments. It is further proved that (1+o(1))inv(3)(n)/n(2) is an element of [112,0.0992) and (1+o(1)) inv(k)( n)/n(2) is an element of [1/2k(k-1)+delta(k),1/2left perpendiculark(2)/2right perpendicular-epsilon(k)], where epsilon(k)>0 for all k >= 3 and delta(k)>0 for all k >= k(0).

It is proved that for integers b, r such that 3 ≤ b < r ≤( [ b+1; 2 ]) - 1 , there exists a red/blue edge-colored graph such that the red degree of every vertex is r, the blue degree of every vertex is b, yet in the closed neighbourhood of every vertex there are more blue edges than red edges. The upper bound r ≤( [ b+1; 2 ]) -1 is best possible for any b ≥ 3 . We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers r, t such that 0 ≤ t ≤r^2/2 - 5r^3/2 , there exists an r-regular graph in which each open neighbourhood induces precisely t edges. Several explicit constructions are introduced and relationships with constant linked graphs, (r, b)-regular graphs and vertex transitive graphs are revealed.

An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let k,ℓ be positive integers. For a tournament T, let fT(k) be the largest integer such that every k-edge coloring of T has a path-monochromatic subtree with at least fT(k) vertices and let fT(k,ℓ) be the restriction to subtrees of depth at most ℓ. It was proved by Landau that fT(1,2)=n and proved by Sands et al. that fT(2)=n where |V(T)|=n. Here we consider fT(k) and fT(k,ℓ) in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-Häggkvist Conjecture. We also study the typical value of fT(k) and fT(k,ℓ), i.e., when T is a random tournament.

For integers $k,n$ with $1 \le k \le n/2$, let $f(k,n)$ be the smallest integer $t$ such that every $t$-connected $n$-vertex graph has a spanning bipartite $k$-connected subgraph. A conjecture of Thomassen asserts that $f(k,n)$ is upper bounded by some function of $k$. The best upper bound for $f(k,n)$ is by Delcourt and Ferber who proved that $f(k,n) \le 10^{10}k^3 \log n$. Here it is proved that $f(k,n) \le 22k^2 \log n$. For larger $k$, stronger bounds hold. In the linear regime, it is proved that for any $0 < c < \frac{1}{2}$ and all sufficiently large $n$, $f(\lfloor cn \rfloor, n) \le 30\sqrt{c}n$. In the polynomial regime, it is proved that for any $\frac{1}{3} \le \alpha < 1$ and all sufficiently large $n$, $f(\lfloor n^\alpha \rfloor ,n) \le 9n^{(1+\alpha)/2}$.

For every fixed $k \ge 4$, it is proved that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed $k$-cycles, then there exists a set of at most $\frac{2}{3}kt+ o(n^2)$ arcs that meets all directed $k$-cycles and that the set of $k$-cycles admits a fractional cover of value at most $\frac{2}{3}kt$. It is also proved that the ratio $\frac{2}{3}k$ cannot be improved to a constant smaller than $\frac{k}{2}$. For $k=5$ the constant $2k/3$ is improved to $25/8$ and for $k=3$ it was recently shown by Cooper et al. that the constant can be taken to be $9/5$. The result implies a deterministic polynomial time $\frac{2}{3}k$-approximation algorithm for the directed $k$-cycle cover problem, improving upon a previous $(k{-}1)$-approximation algorithm of Kortsarz et al. More generally, for every directed graph $H$ we introduce a graph parameter $f(H)$ for which it is proved that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint $H$-copies, then there exists a set of at most $f(H)t+ o(n^2)$ arcs that meets all $H$-copies and that the set of $H$-copies admits a fractional cover of value at most $f(H)t$. It is shown that for almost all $H$ it holds that $f(H) \approx |E(H)|/2$ and that for every $k$-vertex tournament $H$ it holds that $f(H) \le \lfloor k^2/4 \rfloor$.

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N. The largest value of s(H) over all hypergraphs on n vertices and m hyperedges is denoted s(n,m). We prove that s(n,m) is almost-quadratic in m as long as m is not too large. More precisely, the following holds: If n < m < n^O(1) then s(n,m)= m^2/w(m), where w(m) is a function that goes to infinity and is smaller than any polynomial in m. The parameter s(n,m) has close connections to several other graph and hypergraph functions, such as the irregularity strength of hypergraphs. Our result has several applications, notably: 1. We answer a question of Gyarfas et al. whether there are n-vertex hypergraphs with irregularity strength greater than 2n. In fact we show that there are n-vertex hypergraphs with irregularity strength at least n^2-o(1). 2. Our results imply that s*(n)=n^2/w(n) where s*(n) is the distinguishing closed-neighborhood number, i.e., the smallest integer N such that any n-vertex graph allows for a vertex labeling with positive integers at most N so that the sums of labels on distinct closed neighborhoods of vertices are distinct.

In a recent breakthrough, Gilmer proved the union closed conjecture up to a constant factor. Using Gilmer's method and additional ideas, Chase and Lovett proved an optimal result for almost union-closed set systems. Here that result is extended to higher order unions.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time Õ(ndk), where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time Õ(mdk). Conversely, one can transform any O(nbk) algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an Õ(mbk) time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

For a graph G, let c(k)(G) be the number of spanning trees of G with maximum degree at most k. For k >= 3, it is proved that every connected n-vertex r-regular graph G with r >= 12 n/k+1 satisfies c(k)(G)(1/n) >= (1 - o(n)(1))r . z(k), where z(k) > 0 approaches 1 extremely fast (e.g., z(10) = 0.999971). The minimum degree requirement is essentially tight as for every k >= 2 there are connected n-vertex r-regular graphs G with r = left perpendicularn /(k + 1)right perpendicular - 2 for which c(k)(G) = 0. Regularity may be relaxed, replacing r with the geometric mean of the degree sequence and replacing z(k) with z(k)* > 0 that also approaches 1, as long as the maximum degree is at most n(1 - (3 + o(k)(1))root ink/k). The same holds with no restriction on the maximum degree as long as the minimum degree is at least n/k(1 + o(k)(1)).

Let T be an arbitrary phylogenetic tree with n leaves. It is well known that the average quartet distance between two assignments of taxa to the leaves of T is 2/3(n/4). However, a longstanding conjecture of Bandelt and Dress asserts that (23+o(1))(n/4) is also the maximum quartet distance between two assignments. While Alon, Naves, and Sudakov have shown this indeed holds for caterpillar trees, the general case of the conjecture is still unresolved. A natural extension is when partial information is given: the two assignments are known to coincide on a given subset of taxa. The partial information setting is biologically relevant as the location of some taxa (species) in the phylogenetic tree may be known, and for other taxa it might not be known. What can we then say about the average and maximum quartet distance in this more general setting? Surprisingly, even determining the average quartet distance becomes a nontrivial task in the partial information setting and determining the maximum quartet distance is even more challenging, as these turn out to be dependent on the structure of T. In this paper we prove nontrivial asymptotic bounds that are sometimes tight for the average quartet distance in the partial information setting. We also show that the Bandelt and Dress conjecture does not generally hold under the partial information setting. Specifically, we prove that there are cases where the average and maximum quartet distance substantially differ.

Let $H$ be a directed acyclic graph (dag) that is not a rooted star. It is known that there are constants $c=c(H)$ and $C=C(H)$ such that the following holds for $D_n$, the complete directed graph on $n$ vertices. There is a set of at most $C\log n$ directed acyclic subgraphs of $D_n$ that covers every $H$-copy of $D_n$, while every set of at most $c\log n$ directed acyclic subgraphs of $D_n$ does not cover all $H$-copies. Here this dichotomy is considerably strengthened. Let ${\vec G}(n,p)$ denote the probability space of all directed graphs with $n$ vertices and with edge probability $p$. The fractional arboricity of $H$ is $a(H) = max \{\frac{|E(H')|}{|V(H')|-1}\}$, where the maximum is over all non-singleton subgraphs of $H$. If $a(H) = \frac{|E(H)|}{|V(H)|-1}$ then $H$ is totally balanced. Complete graphs, complete multipartite graphs, cycles, trees, and, in fact, almost all graphs, are totally balanced. It is proven that: Let $H$ be a dag with $h$ vertices and $m$ edges which is not a rooted star. For every $a^* > a(H)$ there exists $c^* = c^*(a^*,H) > 0$ such a.a.s. $G \sim {\vec G}(n,n^{-1/a^*})$ has the property that every set $X$ of at most $c^*\log n$ directed acyclic subgraphs of $G$ does not cover all $H$-copies of $G$. Moreover, there exists $s(H) = m/2 + O(m^{4/5}h^{1/5})$ such that the following stronger assertion holds for any such $X$: there is an $H$-copy in $G$ that has no more than $s(H)$ of its edges covered by each element of $X$. If $H$ is totally balanced then for every $0 < a^* < a(H)$, a.a.s. $G \sim {\vec G}(n,n^{-1/a^*})$ has a single directed acyclic subgraph that covers all its $H$-copies. As for the first result, note that if $h=o(m)$ then $s(H)=(1+o_m(1))m/2$ is about half of the edges of $H$. In fact, for infinitely many $H$ it holds that $s(H)=m/2$, optimally. As for the second result, the requirement that $H$ is totally balanced cannot, generally, be relaxed.