arXiv subjects

Bernard Lidický

Publications and source records attributed to Bernard Lidický.

At least 19 recordsLinked to original sources

arXiv2026

The Spectrum of Triangle-free Graphs

Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \leq \frac{15n}{94}< \frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of Erdős: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $\left\lfloor\frac{n}{2}\right\rfloor$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.

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arXiv2026

Maximizing $K_r + I_r$ in graphs with fixed edge density

For every integer $r\ge4$, and $ρ\in [0,1]$, we asymptotically determine the maximum proportion of $r$-element sets of vertices that induce either a clique or an independent set in a large graph with density $ρ$. This generalizes a result of Olpp for $r=3$. After the initial idea for the main proof was found by the authors, various AI models were used to streamline the argument and perform the calculations necessary for completion of the proof.

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arXiv2026

Sidorenko property and forcing in regular tournaments

We give a complete characterization of tournaments H that have the Sidorenko property with respect to nearly regular tournaments, i.e., the homomorphism density of H among all nearly regular tournaments is minimized by a random tournament. Corollaries of our result are a positive answer to the question of Noel, Ranganathan and Simbaqueba whether there exist infinitely many non-transitive tournaments that are quasirandom forcing for nearly regular tournaments, and a negative answer to their question whether almost every tournament is quasirandom forcing for nearly regular tournaments.

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arXiv2026

Density of rainbow triangles and properly colored $K_4$'s

We establish a sharp upper bound on the number of properly $3$-edge-colored $K_4$'s in graphs with $R$ red, $G$ green and $B$ blue edges. We give a computer-free flag-algebra proof of this bound, and we also convert our proof into a classical counting proof and an entropy proof. Additionally, for every $k\ge 4$, for a fixed rainbow coloring $F$ of a complete graph $K_k$, we give a sharp upper bound on the number copies of $F$ in a $\binom{k}{2}$-edge-colored graph. Our proof of this result relies on a new flag-algebra version of Hölder's inequality. We also give a computer-free flag-algebra proof of the fact that a graph with $R$ red, $G$ green, and $B$ blue edges has at most $\sqrt{2 RGB}$ rainbow triangles, which was originally proven by T.-W. Chao and H.-H. H. Yu using the entropy method. We also provide an even shorter entropy proof of their result.

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arXiv2026

Local maximum of inducibility profiles

For a graph $G$ and $e\in [0,1]$, denote by $I_G(e)$ the supremum of induced density of $G$ over $n$-vertex graphs with edge density $e$ as $n$ goes to infinity. Liu, Mubayi and Reiher asked if there exists a graph $G$, where $I_G(e)$ has a non-trivial local maximum. In this paper, we answer this problem affirmatively. We first show that $I_{K_{2,2,1}}(e)$ has at least two local maxima in $(0,1)$. Part of this proof is using flag algebras. Additionally, we determine $I_{K_{2,2,1}}(e)$, when $e=(k-1)/k$ for every integer $k\ge 3.$ We also prove that $I_{K_t^-}(e)$ has a non-global local maximum for every $t\in\{5,8,11,\ldots,$ $74\}$. The proof combines a symmetrization theorem of Schelp and Thomason with Reiher's clique density theorem.

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arXiv2026

Bipartite cuts in Ramsey-Turán style

We prove that every $K_5$-free $n$-vertex graph with sublinear independence number can be made bipartite by removing at most $n^2(1/18+o(1))$ edges, where the constant $1/18$ is best possible. The proof method is related to extensions of Turán Theorem in edge-weighted settings, and part of the proof uses flag algebra.

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arXiv2026

An automated proof that R(B_8,B_10)=37

We present a short proof that the book Ramsey number $R(B_8,B_{10})$ equals 37. The lower bound $R(B_8,B_{10}) \ge 37$ is already available in the literature, so it is enough to rule out a 37-vertex graph containing neither a copy of $B_8$ nor a copy of $B_{10}$ in its complement. The problem as well as the proof were found with AutoMath, an AI-assisted mathematical discovery workflow developed by the first author. A Lean formalization of the upper-bound argument is available in the accompanying repository.

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arXiv2026

Flow-critical graphs

Lovász et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident edges of a single vertex $z$ with bounded degree. We extend this theorem of Lovász et al. to allow $z$ to have arbitrary degree, but with the additional assumption that there is another vertex $x$ with large degree and no small cut separating $x$ and $z$. Using this theorem, we prove two results regarding the generation of minimal graphs with the property that prescribing the edges incident to a vertex with specific flow does not extend to a nowhere-zero $3$-flow. We use this to further strengthen the theorem of Lovász et al., as well as make progress on a conjecture of Li et al.

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arXiv2026

Semi-Inducibility of some small graphs

Let $H$ be a fixed graph whose edges are colored red and blue and let $β\in [0,1]$. Let $I(H, β)$ be the (asymptotically normalized) maximum number of copies of $H$ in a large red/blue edge-colored complete graph $G$, where the density of red edges in $G$ is $β$. This refines the problem of determining the semi-inducibility of $H$, which is itself a generalization of the classical question of determining the inducibility of $H$. The function $I(H, β)$ for $β\in [0,1]$ was not known for any graph $H$ on more than three vertices, except when $H$ is a monochromatic clique (Kruskal-Katona) or a monochromatic star (Reiher-Wagner). We obtain sharp results for some four and five vertex graphs, addressing several recent questions posed by various authors. We also obtain some general results for trees and stars. Many open problems remain.

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arXiv2025

Lower and Upper Bounds for Small Canonical and Ordered Ramsey Numbers

In this paper, we investigate three extensions of Ramsey numbers to other combinatorial settings. We first consider ordered Ramsey numbers. Here, we ask for a monochromatic copy of a linearly ordered graph $G$ in every $2$-edge-coloring of a linearly ordered complete graph $K_n$. The smallest such $n$ is denoted by $\vec{R}(G)$. Next, we study canonical Ramsey numbers. A canonical coloring of a linearly ordered graph $G$ is an edge-coloring in which $G$ is monochromatic, rainbow, or min/max-lexicographic. In the latter case, each pair of edges receives the same color if and only if they share the same first (respectively, second) vertex. Erdős and Rado showed that for every $p$ there exists $n$ such that every edge-coloring of a linearly ordered $K_n$ contains a canonical copy of $K_p$; the smallest such $n$ is denoted by $ER(G)$. Finally, we examine unordered canonical Ramsey numbers, introduced by Richer. An edge-coloring of $G$ is orderable if there exists a linear ordering of its vertices such that the color of each edge is determined by its first vertex. Unlike lexicographic colorings, this notion also includes monochromatic colorings. Richer proved that for all $s$ and $t$, there exists $n$ such that every edge-coloring of $K_n$ contains an orderable copy of $K_s$ or a rainbow $K_t$. The smallest such $n$ is denoted by $CR(s,t)$. In all three settings, we focus on determining the corresponding Ramsey numbers for small graphs $G$. We use tabu search and integer programming to obtain lower bounds, and flag algebras or integer programming to establish upper bounds. Among other results, we determine $\vec{R}(G)$ for all graphs $G$ on up to four vertices except $K_4^-$, $ER(P_4)$ for all orderings of $P_4$, and the exact values $CR(6,3)=26$ and $CR(3,5)=13$.

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arXiv2025

On Relative Ordered Turán Density

For an ordered graph $F$, denote the Turán density by $\vecπ(F)$. The relative Turán density, denoted by $ρ(F)$, is the supremum over $α\in [0,1]$ such that every ordered graph $G$ contains an $F$-free subgraph $G'$ with $e(G') \geq αe(G)$. Reiher, Rödl, Sales and Schacht showed that $ρ(P) = \vecπ(P)/2$ and $ρ(K) = \vecπ(K)$ for any ascending path $P$ or clique $K$. They asked if there are any ordered graphs $F$ with $\vecπ(F)/2 < ρ(F) < \vecπ(F)$. We answer this question in the affirmative by describing a family of such $F$. We also show that the relative Turán densities of a large family of ordered matchings (including $\{\{1,6\}, \{2,3\}, \{4,5\}\}$ and $\{\{1,3\}, \{2,5\}, \{4,6\}\}$) are $0$.

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arXiv2025

Decomposing random regular graphs into stars

We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all $d,k$ such that $d/2 < k \leq d/2 + \max\{1,\frac{1}{6}\log d\}$. More generally, we give a sufficient existence condition that can be checked numerically for any given values of $d$ and $k$. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for $k$-star decompositions in $\mathcal{G}_{n,d}$ for all $d\leq 100$ and $k > d/2$. For smaller values of $k$, the connection between $k$-star decompositions and $β$-orientations allows us to apply results of Thomassen (2012) and Lovász, Thomassen, Wu and Zhang (2013). We prove that random $d$-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of $k$-star decompositions (i) when $2k^2+k\leq d$, and (ii) when $k$ is odd and $k < d/2$.

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arXiv2025

On a conjecture concerning 4-coloring of graphs with one crossing

We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at most 28 vertices, explore the consequences of this conjecture and provide some insights on how it could be proved.

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arXiv2025

11/4-colorability of subcubic triangle-free graphs

We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.

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arXiv2025

Hypergraph Turán Problems in $\ell_2$-Norm

There are various different notions measuring extremality of hypergraphs. In this survey we compare the recently introduced notion of the codegree squared extremal function with the Turán function, the minimum codegree threshold and the uniform Turán density. The codegree squared sum $\textrm{co}_2(G)$ of a $3$-uniform hypergraph $G$ is defined to be the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $x,y$. In other words, this is the square of the $\ell_2$-norm of the codegree vector. We are interested in how large $\textrm{co}_2(G)$ can be if we require $G$ to be $H$-free for some $3$-uniform hypergraph $H$. This maximum value of $\textrm{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$ is called the codegree squared extremal function, which we denote by $\textrm{exco}_2(n,H)$. We systemically study the extremal codegree squared sum of various $3$-uniform hypergraphs using various proof techniques. Some of our proofs rely on the flag algebra method while others use more classical tools such as the stability method. In particular, we (asymptotically) determine the codegree squared extremal numbers of matchings, stars, paths, cycles, and $F_5$, the $5$-vertex hypergraph with edge set $\{123,124,345\}$. Additionally, our paper has a survey format, as we state several conjectures and give an overview of Turán densities, minimum codegree thresholds and codegree squared extremal numbers of popular hypergraphs. We intend to update the arXiv version of this paper regularly.

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arXiv2024

Positive co-degree densities and jumps

The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by $δ_{r-1}^+(H)$, is the largest integer $k$ such that for every $(r-1)$-set $S \subset V(H)$, if $S$ is contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. Given a family $\mathcal{F}$ of $r$-graphs, the positive co-degree Turán function $\mathrm{co^+ex}(n,\mathcal{F})$ is the maximum of $δ_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\mathcal{F}$. The positive co-degree density of $\mathcal{F}$ is $γ^+(\mathcal{F}) = \underset{n \rightarrow \infty}{\lim} \frac{\mathrm{co^+ex}(n,\mathcal{F})}{n}.$ While the existence of $γ^+(\mathcal{F})$ is proved for all families $\mathcal{F}$, only few positive co-degree densities are known exactly. For a fixed $r \geq 2$, we call $α\in [0,1]$ an achievable value if there exists a family of $r$-graphs $\mathcal{F}$ with $γ^+(\mathcal{F}) = α$, and call $α$ a jump if for some $δ> 0$, there is no family $\mathcal{F}$ with $γ^+(\mathcal{F}) \in (α, α+ δ)$. Halfpap, Lemons, and Palmer showed that every $α\in [0, \frac{1}{r})$ is a jump. We extend this result by showing that every $α\in [0, \frac{2}{2r -1})$ is a jump. We also show that for $r = 3$, the set of achievable values is infinite, more precisely, $\frac{k-2}{2k-3}$ for every $k \geq 4$ is achievable. Finally, we determine two additional achievable values for $r=3$ using flag algebra calculations.

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arXiv2024

Small Ramsey numbers for books, wheels, and generalizations

In this work, we give several new upper and lower bounds on Ramsey numbers for books and wheels, including a tight upper bound establishing $R(W_5, W_7) = 15$, matching upper and lower bounds giving $R(W_5, W_9) = 18$, $R(B_2, B_8) = 21$, and $R(B_3, B_7) = 20$, and a number of additional tight lower bounds for books. We use a range of different methods: flag algebras, local search, bottom-up generation, and enumeration of polycirculant graphs. We also explore generalized Ramsey numbers using similar methods. Let $GR(r,K_s,t)$ denote the minimum number of vertices $n$ such that any $r$-edge-coloring of $K_n$ has a copy of $K_s$ with at most $t$ colors. We establish $GR(3,K_4,2) = 10, GR(4,K_4,3) = 10$, and some additional bounds.

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arXiv2024

Proper Rainbow Saturation Numbers for Cycles

We say that an edge-coloring of a graph $G$ is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of $G$ receive the same color. Furthermore, given a fixed graph $F$, we say that $G$ is rainbow $F$-saturated if $G$ admits a proper edge-coloring which does not contain any rainbow subgraph isomorphic to $F$, but the addition of any edge to $G$ makes such an edge-coloring impossible. The maximum number of edges in a rainbow $F$-saturated graph is the rainbow Turán number, whose study was initiated in 2007 by Keevash, Mubayi, Sudakov, and Verstraëte. Recently, Bushaw, Johnston, and Rombach introduced study of a corresponding saturation problem, asking for the minimum number of edges in a rainbow $F$-saturated graph. We term this minimum the proper rainbow saturation number of $F$, denoted $\mathrm{sat}^*(n,F)$. We asymptotically determine $\mathrm{sat}^*(n,C_4)$, answering a question of Bushaw, Johnston, and Rombach. We also exhibit constructions which establish upper bounds for $\mathrm{sat}^*(n,C_5)$ and $\mathrm{sat}^*(n,C_6)$.

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