Dejean's conjecture holds for n >= 30

Brouwer's conjecture holds asymptotically almost surely

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible negative weights as well. For graphs with a fixed number of vertices, the result implies that there are constants $C>0$ and $n_{0}$ such that if $n\geq n_{0}$ then among all $2^{n \choose 2}$ graphs with $n$ vertices, at least $\left(1-\exp\left(-Cn\right)\right)2^{n \choose 2}$ graphs satisfy Brouwer's conjecture.

Meyniel's conjecture holds for random d-regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant $C$, the cop number of every connected graph $G$ is at most $C \sqrt{|V(G)|}$. In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graph…

The 1-2-3 Conjecture almost holds for regular graphs

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set $\{1,2,3,4,5\}$. We show that for regular graphs it is sufficient to use weights $1$, $2$, $3$, $4$. Moreover, we prove the conjecture to hold for every $d$-regular graph with $d\geq 10^8$.

Union-closed Sets Conjecture Holds for Height No More Than 3 and Height No Less Than N-1

For each given union-closed family F of n elements and m sets, we discuss the union-closed sets conjecture from height number of the UC family, which is a natural parameter from lattice theory. In this paper, we call it height number of F(n, m), recorded as H(F). we prove that for any given union-closed family F, union-closed sets conjecture holds if its height number H(F) no more than 3 or no less than n-1. Since the height number H(F) is a positive integer which is bounded between 1 to n. As an attempted approach and framework, if we can prove union-closed sets conjecture holds for all possi…

The Merrifield-Simmons conjecture holds for bipartite graphs

Let $G = (V, E)$ be a graph and $σ(G)$ the number of independent (vertex) sets in $G$. Then the Merrifield-Simmons conjecture states that the sign of the term $σ(G_{-u}) \cdot σ(G_{-v}) - σ(G) \cdot σ(G_{-u-v})$ only depends on the parity of the distance of the vertices $u, v \in V$ in $G$. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.

Rota's Basis Conjecture holds asymptotically

Rota's Basis Conjecture is a well known problem from matroid theory, that states that for any collection of $n$ bases in a rank $n$ matroid, it is possible to decompose all the elements into $n$ disjoint rainbow bases. Here an asymptotic version of this is proved. We show that it is possible to find $n-o(n)$ disjoint rainbow independent sets of size $n-o(n)$.

The CLLC conjecture holds for cyclic outer permutations

Recently, Gross et al. posed the LLC conjecture for the locally log-concavity of the genus distribution of every graph, and provided an equivalent combinatorial version, the CLLC conjecture, on the log-concavity of the generating function counting cycles of some permutation compositions. In this paper, we confirm the CLLC conjecture for cyclic permutations, with the aid of Hultman numbers and by applying the Hermite--Biehler theorem on the generating function of Stirling numbers of the first kind. This leads to a further conjecture that every local genus polynomial is real-rooted.

通过 SAT 求解器证明 Norine 猜想在 $n=7$ 时成立

Proving Norine's Conjecture holds for $n=7$ via SAT solvers

针对 $n$ 维立方体图对跖红/蓝边着色中是否必有对跖顶点由单色路径相连的问题，本文使用 SAT 求解器将 Norine 猜想的成立范围从 $n\le 6$ 推进到 $n=7$。

Gamma-Theta猜想对平面图成立

The Gamma-Theta Conjecture holds for planar graphs

本文证明了Gamma-Theta猜想对平面图成立，即当平面图的支配数等于永恒支配数时，它也等于团覆盖数。

A note on Reed's conjecture

In \cite{reed97}, Reed conjectures that the inequality $χ(G) \leq \left \lceil \textstyle {1/2} (ω(G) + Δ(G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From this it follows that the conjecture holds for graphs with $χ(G) > \left \lceil \frac{|G|}{2} \right \rceil$. In addition, the conjecture holds for graphs with $Δ(G) \geq |G| - \sqrt{|G| + 2α(G) + 1}$. In particular, Reed's conjecture holds for graphs with $Δ(G) \geq |G| - \sqrt{|G| + 7}$. Using these results, we proceed to show that if $|G|$ is an even order counterexample…

The union-closed sets conjecture almost holds for almost all random bipartite graphs

Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl's conjecture.

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

The Buratti-Horak-Rosa Conjecture concerns the possible multisets of edge-labels of a Hamiltonian path in the complete graph with vertex labels $0, 1, \ldots, {v-1}$ under a particular induced edge-labeling. The conjecture has been shown to hold when the underlying set of the multiset has size at most~2, is a subset of $\{1,2,3,4\}$ or $\{1,2,3,5\}$, or is $\{1,2,6\}$, $\{1,2,8\}$ or $\{1,4,5\}$, as well as partial results for many other underlying sets. We use the method of growable realizations to show that the conjecture holds for each underlying set $U = \{ x,y,z \}$ when $\max(U) \leq 7$ …

完美1-因子分解猜想渐近成立

The perfect 1-factorisation conjecture holds asymptotically

研究Anton Kotzig关于完全图Kn的完美1-因子分解猜想，通过证明Kn可分解为n - 1个完美匹配，其中(1 - o(1))n个匹配中任意一对都能形成哈密顿圈，得出该猜想渐近成立的结论。

证明Brouwers猜想(BC)对所有顶点数n > n_0的图成立，假设BC对某个n_0在n< n_0时成立

Proof of Brouwers Conjecture (BC) for all graphs with number of vertices n > n_0 assuming that BC holds for n< n_0 for some n_0

在假设小图满足Brouwers猜想的前提下，证明了所有顶点数大于某个固定阈值n_0的图也满足该猜想。

A generalisation of Seymour's second neighbourhood conjecture

In this note we propose a generalisation of Seymour's Second Neighbourhood Conjecture to two directed graphs on a vertex set. We prove that this generalisation holds in the case of tournaments, and we show that a natural strengthening of this conjecture does not hold.

The Merrifield-Simmons conjecture also holds for parity graphs

The Merrifield-Simmons conjectures states a relation between the distance of vertices in a simple graph $G$ and the number of independent sets, denoted as $σ(G)$, in vertex-deleted subgraphs. Namely, that the sign of the term $σ(G_{-u}) \cdot σ(G_{-v}) - σ(G) \cdot σ(G_{-u-v})$ only depends on the parity of the distance of $u$ and $v$ in $G$. We prove this statement in the case of parity graphs and give some evidence that this result may not be further generalized to other classes of graphs.

Enomoto and Ota's conjecture holds for large graphs

In 2000, Enomoto and Ota conjectured that if a graph $G$ satisfies $σ_{2}(G) \geq n + k - 1$, then for any set of $k$ vertices $v_{1}, \dots, v_{k}$ and for any positive integers $n_{1}, \dots, n_{k}$ with $\sum n_{i} = |G|$, there exists a partition of $V(G)$ into $k$ paths $P_{1}, \dots, P_{k}$ such that $v_{i}$ is an end of $P_{i}$ and $|P_{i}| = n_{i}$ for all $i$. We prove this conjecture when $|G|$ is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices.

哈迪格猜想对强单型多面体成立

Hadwiger's conjecture holds for strongly monotypic polytopes

本文证明了强单型多面体满足哈迪格猜想，核心方法是利用多面体的结构特性，主要贡献是扩展了猜想的应用范围。

Bollobás-Nikiforov猜想渐近几乎必然成立

Bollobás-Nikiforov conjecture holds asymptotically almost surely

该研究针对Bollobás-Nikiforov提出的图邻接矩阵特征值与边数、团数关系的猜想，证明了在随机图序列中，当顶点数趋于无穷时，该猜想成立的概率趋于1，即猜想渐近几乎必然成立。

九名跑者持有的孤独跑者猜想

The lonely runner conjecture holds for nine runners

本文证明了九名跑者情况下孤独跑者猜想成立，通过改进原有方法实现了证明。

The Fuglede Conjecture holds in \(\F_p^3\) for p=5,7

For p=5,7, we show that a subset \(E \subset \F_p^3\) is spectral if and only if E tiles \(\F_p^3\) by translation. Additionally, we give an alternate proof that the conjecture holds for p=3.

全染色猜想对不含三种特殊子图的平面图成立

The Total Coloring Conjecture holds for planar graphs without three special subgraphs

本文证明全染色猜想对最大度为六且不含蘑菇、帐篷和锥体三种子结构的平面图成立，从而推广了此前关于禁止4-扇的结果。

The Komlos Conjecture Holds for Vector Colorings

The Komlos conjecture in discrepancy theory states that for some constant K and for any m by n matrix A whose columns lie in the unit ball there exists a +/- 1 vector x such that the infinity norm of Ax is bounded above by K. This conjecture also implies the Beck-Fiala conjecture on the discrepancy of bounded degree hypergraphs. Here we prove a natural relaxation of the Komlos conjecture: if the columns of A are assigned unit real vectors rather than +/- 1 then the Komlos conjecture holds with K=1. Our result rules out the possibility of a counterexample to the conjecture based on semidefinite…

Rota's basis conjecture holds for random bases of vector spaces

In 1989, Rota conjectured that, given $n$ bases $B_1,\dots,B_n$ of the vector space $\mathbb{F}^n$ over some field $\mathbb{F}$, one can always decompose the multi-set $B_1\cup \dots \cup B_n$ into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases $B_1,\dots,B_n$. More specifically, our first result shows that Rota's basis conjecture holds with probability $1-o(1)$ as $n\to \infty$ if the bases $B_1,\dots,B_n$ are chosen independently uniformly at random among all bases of $\mathbb{F}_q^n$ for some finite …

Meyniel's conjecture holds for random graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant $C$, the cop number of every connected graph $G$ is at most $C \sqrt{|V(G)|}$. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. We do this by first showing that the conjecture holds for a general class of graphs with some spe…

Vizing's edge-recoloring conjecture holds

In 1964 Vizing proved that starting from any k-edge-coloring of a graph G one can reach, using only Kempe swaps, a ($Δ$ + 1)-edge-coloring of G where $Δ$ is the maximum degree of G. One year later he conjectured that one can also reach a $Δ$-edge-coloring of G if there exists one. Bonamy et. al proved that the conjecture is true for the case of triangle-free graphs. In this paper we prove the conjecture for all graphs.

Lexicographic shellability, matroids and pure order ideals

In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This suggests that a pure $O$-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank $d$ matroids on at most $2d$ elements, then it holds for all matro…

Fuglede's conjecture holds in $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$

Fuglede's conjecture states that for a subset $Ω$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^{2}(Ω)$ if and only if it tiles the group by translation. In this paper, we prove a divisibility property for a set in $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$. Then using the divisibility property and equi-distributed property, we prove that Fuglede's conjecture holds in the group $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$.

The bunkbed conjecture holds in the $p\uparrow 1$ limit

Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G \times K_2$, which has vertex set $V\times \{0,1\}$ with "horizontal'' edges inherited from $G$ and additional "vertical'' edges connecting $(w,0)$ and $(w,1)$ for each $w \in V$. Kasteleyn's bunkbed conjecture states that for each $u,v \in V$ and $p\in [0,1]$, the vertex $(u,0)$ is at least as likely to be connected to $(v,0)$ as to $(v,1)$ under Bernoulli-$p$ bond percolation on the bunkbed graph. We prove that the conjecture holds in the $p \uparrow 1$ limit in the sense that for each finite graph $G$ there…

Catlin's conjecture and maximum eulerian subgraph

A graph $G=(V(G), E(G))$ is supereulerian if it has a spanning Eulerian subgraph. Let $\ell(G)$ be the maximum number of edges of spanning Eulerian subgraphs of a supereulerian graph $G$. In $1996$, Catlin conjectured that if $G$ is a supereulerian graph, then $\ell(G)\ge \frac{2}{3}|E(G)|$. But in $2004$, infinitely many counterexamples were found for this conjecture and it was shown that this conjecture holds for $r$-regular graphs when $r\neq 5$. In this paper we show that Catlin's Conjecture holds for graphs having no vertex with degree $3$ and also it holds for $5$-regular graphs. Moreove…

燃烧数猜想对阶为$n$且2度顶点数至多为$\u230a\sqrt{n-1}\u230b$的树成立

The burning number conjecture holds for trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices

本文围绕图的燃烧数猜想展开，推导得出含$n_2$个2度顶点的$n$阶树的燃烧数上界公式，进而证实该猜想对2度顶点数不超过$\lfloor\sqrt{n-1}\rfloor$的$n$阶树成立。

The Fuglede conjecture holds in $\mathbb{Z}^3_5$

The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by T. Tao for dimensions 5 and higher by giving a counterexample in $\mathbb{Z}_3^5$. We present a computer program that determines that the Fuglede conjecture holds in $\mathbb{Z}_5^3$ by exhausting the search space. A. Iosevich, A. Mayeli and J. Pakianathan showed that the Fuglede conjecture holds over prime fields when the dimension does not exceed 2. The question for dimension 3 was previously addressed by Aten et al. for $p=3$. In this paper we build upon the results o…

Seymour 和 Woodall 的猜想对独立数为二的图成立

Seymour and Woodall's conjecture holds for graphs with independence number two

本文针对独立数为二的图，证明其在 $2\ell\leqχ(G)$ 时包含 $K^{\ell}_{\ell,χ(G)-\ell}$-minor，从而验证 Seymour–Woodall 猜想在该图类中成立。

An Improved Bound for the Linear Arboricity Conjecture

In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph $G$ of maximum degree $Δ$ can be decomposed into at most $\left\lceil \fracΔ{2}\right\rceil$ linear forests. (A forest is linear if all of its components are paths.) In 1988, Alon proved the conjecture holds asymptotically. The current best bound is due to Ferber, Fox and Jain from 2020 who showed that $\fracΔ{2}+ O(Δ^{.661})$ suffices for large enough $Δ$. Here, we show that $G$ admits a decomposition into at most $\fracΔ{2}+ 3\sqrtΔ \log^4 Δ$ linear forests provided $Δ$ is large enough. Mor…

The Hirsch conjecture holds for normal flag complexes

Using an intuition from metric geometry, we prove that any flag and normal simplicial complex satisfies the non-revisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus the dimension, as in the Hirsch conjecture. This proves the Hirsch conjecture for all flag polytopes, and more generally, for all (connected) flag homology manifolds.

关于Fan的4-流猜想

On Fan's conjecture about $4$-flow

本文证明了Fan关于无桥图中4-流与回路覆盖的猜想在回路长度不超过35时成立，推广了先前结果。

The "Most informative boolean function" conjecture holds for high noise

We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $ε\ge 1/2 - δ$, for some absolute constant $δ> 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ε$, then, provided $ε\ge 1/2 - δ$, for any boolean function $f$ holds $I(f(X);Y) \le 1 - H(ε)$. This conjecture was previously known to hold only for balanced functions.

孤独跑者猜想对八名跑者成立

The lonely runner conjecture holds for eight runners

本文通过计算机验证与最小反例规模界，证明了孤独跑者猜想对八名跑者成立，并指出方法可推广至更多跑者。

The Tu--Deng Conjecture holds almost surely

The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\leq t<2^k-1$. Then \[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leq 2^{k-1}.\] We prove that the Tu--Deng Conjecture holds almost surely in the following sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\rightarrow\infty$. Moreover, we prove that the Tu--Deng Conjecture i…

Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two

The vertex (resp. edge) metric dimension of a connected graph G; denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S in V (G) which distinguishes all pairs of vertices (resp. edges) in G: Bounds dim(G) <= L(G)+2c(G) and edim(G) <= L(G) + 2c(G); where c(G) is the cyclomatic number in G and L(G) depends on the number of leaves in G, are known to hold for cacti and are conjectured to hold for general graphs. In leafless graphs it holds that L(G) = 0; so for such graphs the conjectured upper bound becomes 2c(G). In this paper, we show that the bound 2c(G) cannot be attain…

Novák's conjecture on cyclic Steiner triple systems and its generalization

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic $(v,k,λ)$-designs with $1 \leq λ\leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime a…

The 1-2-3 Conjecture holds for graphs with large enough minimum degree

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a universal constant $K$, say $K=3$, such that one may always dispose of such pairs from any given connected graph with at least three vertices by blowing its selected edges into at most $K$ parallel edges? This question was first posed in 2004 by Karoński, Łuczak and Thomason, who equivalently asked if one may assign weights $1,2,3$ to the edges of every such graph s…

Conjecture $\mathcal{O}$ holds for the odd symplectic Grassmannian

Let $\mathrm{IG}(k, 2n+1)$ be the odd-symplectic Grassmannian. Property $\mathcal{O}$, introduced by Galkin, Golyshev and Iritani for arbitrary complex, Fano manifolds $X$, is a statement about the eigenvalues of the linear operator obtained by the quantum multiplication by the anticanonical class of $X$. We prove that property $\mathcal{O}$ holds in the case when $X= \mathrm{IG}(k, 2n+1)$ is an odd-symplectic Grassmannian. The proof uses the combinatorics of the recently found quantum Chevalley formula for $\mathrm{IG}(k, 2n+1)$, together with the Perron-Frobenius theory of nonnegative matric…

燃烧数猜想渐近成立

The Burning Number Conjecture Holds Asymptotically

本文证明了燃烧数猜想渐近成立，即对于任意$n$个顶点的图$G$，其燃烧数$b(G)$不超过$(1+o(1))\sqrt n$，从而在渐近意义上解决了该猜想。

Markoff单射猜想的一个$q$-模拟成立

A $q$-analog of the Markoff injectivity conjecture holds

该研究证明Markoff单射猜想的$q$-模拟在Christoffel词上成立，基于$q=\text{exp}(iπ/3)$赋值完成证明，还将问题推广至任意词并给出不满足单射性的词对族。

The PFR Conjecture Holds for Two Opposing Special Cases

Let $A \subseteq F_2^n$ be a set with $|2A| = K|A|$. We prove that if (1) for at least a fraction $1-K^{-9}$ of all $s \in 2A$, the set $(A+s) \cap A$ has size at most $L\cdot|A|/K$, or (2) for at least a fraction $K^{-L}$ of all $s \in 2A$, the set $(A+s) \cap A$ has size at least $|A|\cdot(1- K^{-1/L})$, then there is a subset $B \subseteq A$ of size $|A|/K^{O_L(1)}$ such that $\mathrm{span}(B) \leq K^{O_L(1)}\cdot|A|$.

Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere

Frankl and Füredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realised by the initial segment of the colexicographic order. In particular, in the principal case $m=\binom{t}{r}$ their conjecture states that every $H\subseteq \mathbb{N}^{(r)}$ of size $\binom{t}{r}$ satisfies \begin{align*} \max \{\sum_{A \in H}\prod_{i\in A} y_i \ \colon \ y_1,y_2,\ldots \geq 0; \sum_{i\in \mathbb{N}} y_i=1 \}&\leq \frac{1}{t^r}\binom{t}{r}. \end{align*} We prove the above statement for all $r\geq 4$ and large values of $t$ (the case $r=3$ was settled by …

Voronoi's Conjecture for extensions of Voronoi parallelohedra

Let $I$ be a segment in the $d$-dimensional Euclidean space $\mathbb E^d$. Let $P$ and $P+I$ be parallelohedra in $\mathbb E^d$, where "+" denotes the Minkowski sum. We prove that Voronoi's Conjecture holds for $P+I$, i.e. $P+I$ is a Voronoi parallelohedron for some Euclidean metric in $\mathbb E^d$, if Voronoi's Conjecture holds for $P$.

Lovász猜想在中等密度Cayley图中成立

The Lovász conjecture holds for moderately dense Cayley graphs

研究证明了大连接Cayley图存在哈密顿回路，推进了Lovász猜想，并改进了之前的结果。

Nivat's conjecture holds for sums of two periodic configurations

Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps $\mathbb Z^2 \rightarrow \mathcal A$ where $\mathcal A$ is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let $P_c(m,n)$ denote the number of distinct $m \times n$ block patterns occurring in a configuration $c$. Configurations satisfying $P_c(m,n) \leq mn$ for some $m,n \in \mathbb N$ are said to have low rectangular complexity. Nivat conjectured that such configurations are necessarily periodic. Recently,…

Aharoni的彩虹圈猜想在相差一个加性常数的意义下成立

Aharoni's rainbow cycle conjecture holds up to an additive constant

本研究针对Aharoni提出的彩虹圈猜想，证明对于固定r，该猜想在相差一个O(r^5 log²r)阶加性常数的意义下成立，推进了边着色图中彩虹圈存在性问题的研究。

Wilf's conjecture in fixed multiplicity

We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's conjecture holds whenever $m \le 18$. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf's conjecture via a combinatorially-flavored game played on the elements of a certain finite poset.

Equitable vertex arboricity conjecture holds for graphs with low degeneracy

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every $d$-degenerate graph with maximum degree at most $Δ$ is equitably tree-$k$-colorable for every integer $k\geq (Δ+1)/2$ provided that $Δ\geq 9.818…

The Fuglede Conjecture holds in ${\Bbb Z}_p \times {\Bbb Z}_p$

In this paper we study subsets $E$ of ${\Bbb Z}_p^d$ such that any function $f: E \to {\Bbb C}$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in ${\Bbb Z}_p^2$ which says that $E \subset {\Bbb Z}_p^2$ is spectral if and only if $E$ tiles ${\Bbb Z}_p^2$ by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.