Largest Integer Research Articles

The Maximum Zero-Sum Partition problem

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a1,a2,…,an} of integers ai∈Z⁎ (where Z⁎ denotes the set of non-zero integers) such that ∑i=1nai=0, find a maximum cardinality partition {S1,S2,…,Sk} of S such that, for every 1≤i≤k, ∑aj∈Siaj=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.

Path-monochromatic bounded depth rooted trees in (random) tournaments

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.

Relating the annihilation number and the 2-domination number for unicyclic graphs

The 2-domination number ?2(G) of a graph G is the minimum cardinality of a set S ? V(G) such that every vertex from V(G)\S is adjacent to at least two vertices in S. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of its edges. It was conjectured that ?2(G) ? a(G)+1 holds for every non-trivial connected graph G. The conjecture was earlier confirmed for graphs of minimum degree 3, trees, block graphs and some bipartite cacti. However, a class of cacti were found as counterexample graphs recently by Yue et al. [9] to the above conjecture. In this paper, we consider the above conjecture from the positive side and prove that this conjecture holds for all unicyclic graphs.

Results on Grundy Chromatic Number of Join Graph of Graphs

A Grundy k -coloring of a graph G is a proper k -coloring of vertices in G using colors { 1 , 2 , ⋯ , k } such that for any two colors x and y , x < y , any vertex colored y is adjacent to some vertex colored x . The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph G , denoted by Γ ( G ) , is the largest integer k , such that there exists a Grundy k -coloring for G . It can be easily seen that Γ ( G ) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of G . In this paper, we obtain the Grundy chromatic number of Cartesian Product of path graph, complete graph, cycle graph, complete graph, wheel graph and star graph.

Counterexamples to the Characterisation of Graphs with Equal Independence and Annihilation Number

In this paper, we disprove the claimed characterisation of graphs with equal independence and annihilation number as proposed by Larson and Pepper [Electron. J. Comb. 2011]. The annihilation number of a graph is defined as the largest integer $p$ such that the sum of its smallest $p$ degrees is greater than or equal to its size, i.e., its number of edges. Larson and Pepper claimed that for a given graph $G=(V,E)$, its independence number $\alpha(G)$ equals its annihilation number $a(G)$ if and only if $$\begin{array}{ll}(1)~~ a(G)\geq \frac n2:& \alpha'(G)=a(G)\\[2mm](2)~~ a(G)= \frac{n-1}{2}:& \alpha'(G-v)=a(G) ~\text{ for some } v\in V.\end{array}$$This paper provides series of counterexamples with an arbitrarily large number of vertices, an arbitrarily large number of components, an arbitrarily large independence number, and an arbitrarily large difference between the critical and the regular independence number. Furthermore, we identify the error in the proof of Larson and Pepper's theorem. Yet, we show that the theorem still holds for bipartite graphs and connected claw-free graphs.

The Generalized Frobenius Problem via Restricted Partition Functions

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give formulas for these values by understanding the level sets of the restricted partition function (the function $f(t)$ giving the number of representations of $t$). Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.

P-Numerical semigroups with p-symmetric properties

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation [Formula: see text] ([Formula: see text] are given positive integers with [Formula: see text]) does not have a non-negative integer solution [Formula: see text]. The generalized Frobenius number (called the [Formula: see text]-Frobenius number) is the largest integer such that this linear equation has at most [Formula: see text] solutions. That is, when [Formula: see text], the [Formula: see text]-Frobenius number is the original Frobenius number. In this paper, we introduce and discuss [Formula: see text]-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer [Formula: see text], [Formula: see text]-gaps, [Formula: see text]-symmetric semigroups, [Formula: see text]-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When [Formula: see text], they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

Hat Guessing Numbers of Strongly Degenerate Graphs

Assume players are placed on vertices of a graph . The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a color out of available colors. The players can see the hat of each of their neighbors in but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer such that there is a winning strategy for the players is denoted by , and this is called the hat guessing number of . Although this game has received much attention in recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from to 40 and further derive upper bounds on the hat guessing number for any class of -free graphs with bounded expansion, such as the class of -free planar graphs; more generally, for -free graphs with bounded Hadwiger number or without a -subdivision; and for Erdős–Rényi random graphs with constant average degree.

Bounds for invariants of numerical semigroups and Wilf’s conjecture

Given coprime positive integers g_1< ldots < g_e, the Frobenius number F=F(g_1,ldots ,g_e) is the largest integer not representable as a linear combination of g_1,ldots ,g_e with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F+1 le e n. We provide bounds for g_1 and for the type of the numerical semigroup S=langle g_1,ldots ,g_e rangle in function of e and n, and use these bounds to prove that F+1 le q e n, where q= Bigg lceil frac{F+1}{g_1} Bigg rceil , and F+1 le e n^2. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup S=langle g_1,ldots ,g_e rangle is almost-symmetric.

On t-Intersecting Hypergraphs with Minimum Positive Codegrees

For a hypergraph , define the minimum positive -degree to be the largest integer such that every -set which is contained in at least one edge of is contained in at least edges. For and , we prove that for -vertex -intersecting -graphs with , the unique hypergraph with the maximum number of edges is the hypergraph consisting of every edge which intersects a set of size in at least vertices provided is sufficiently large. This generalizes work of Balogh, Lemons, and Palmer who proved this for , as well as the Erdős-Ko-Rado theorem when .

Nilpotent Lie algebras in which all proper subalgebras have class at most n, II

Let be a field and let L be a finitely generated nilpotent Lie -algebra of class (exactly) c. Let n be the largest integer such that L has a proper subalgebra of class n, and let be the smallest integer such that L can be generated by d elements. In this work, we suppose that . We show that if , then either and L is the unique non-abelian nilpotent Lie -algebra of dimension 3, or the characteristic of is 3, and L is a certain Lie -algebra of dimension 7 in which every maximal subalgera has class 2. We compare this with previous results that deal with the case .

The Saturation Spectrum for Antichains of Subsets

Extending a classical theorem of Sperner, we characterize the integers m such that there exists a maximal antichain of size m in the Boolean lattice Bn, that is, the power set of $[n]:=\{1,2,\dots ,n\}$ , ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family $\mathcal F$ of k-sets with $|{\mathcal {F}}|=t$ such that the shadow of $\mathcal {F}$ has size s, where the shadow of $\mathcal F$ is the collection of (k − 1)-sets that are contained in at least one member of $\mathcal F$ . We provide a complete answer for $t\leqslant k+1$ . Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is $\sqrt 2k^{3/2}+\sqrt [4]{8}k^{5/4}+O(k)$ .

Common edge independence number of a tree (Brief Announcement)

The cardinality of a largest matching of G, denoted by α'(G), is called the upper matching number of G. The lower matching number i'(G) of a graph G is the cardinality of a smallest maximal matching of G. We introduce the concept of the common edge independence number of a graph G, denoted by α'c(G), is the largest integer k such that every edge of G belongs to a matching that has at least k edges. For any graph G, the relations between above parameters are given by the chain of inequalities i'(G)≤α'c(G)≤α'(G). We study relations between this three parameters, in particular we show that the difference between α'c(G) and i'(G) can be arbitrarily large while α'(G) and α'c(G) may differ by at most one. We also characterize the trees T for which i'(T)=α'c(T), and the trees T for which α'c(T)=α'(T).

On a conjecture of Ramírez Alfonsín and Skałba

Let 1⩽a<b be two relatively prime integers. Sylvester found that ab−a−b is the largest integer which can not be represented by ax+by(x,y∈Z⩾0) about 160 years ago and this number shall be denoted by ga,b. Let N(a,b)={n:n⩽ga,b,n=ax+by,x,y∈Z⩾0} and πa,b be the number of primes in N(a,b). Recently, Ramírez Alfonsín and Skałba proved thatπa,b≫εga,b(log⁡ga,b)2+ε for any fixed ε>0. They further conjectured that the order of the magnitude of πa,b is 12π(ga,b), where π(x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a,b with 1⩽a<b and (a,b)=1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem.

Distinct angles in general position

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is the Erdős distinct angle problem, the problem of finding the minimum number of distinct angles between n non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by analogous questions in the distance setting.In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by n points in general position from O(nlog2⁡(7)) to O(n2). We consider a point-set to be in general position if no three points lie on a common line and no four lie on a common circle. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we introduce a construction employing the geometric properties of a logarithmic spiral, sidestepping the need for a projection.We also apply this configuration to reduce the upper bound on the largest integer such that any set of n points in general position has a subset of that size with all distinct angles. This bound is decreased from O(nlog2⁡(7)/3) to O(n1/2).

Complete family reduction and spanning connectivity in line graphs

Complete families of connected graphs, introduced by Catlin in the 1980s, have been known useful in the study of certain graphical properties that are closed under taking contractions. We show that given any complete family C of connected graphs such that C contains graphs with sufficiently many edge-disjoint spanning trees, for any real number a and b with 0<a<1, there exists a finite obstacle family F=F(a,b,C) such that for any simple graph G on n vertices satisfying the Ore-type degree conditionmin⁡{dG(u)+dG(v):u,v∈V(G) and uv∉E(G)}≥an+b, either G∈C or G can be contracted to a member in F. This result is applied to the study of spanning connectivity of line graphs. The spanning connectivity is the largest integer s such that for any k with 0≤k≤s and for any u,v∈V(G) with u≠v, G has a spanning subgraph H consisting of k internally disjoint (u,v)-paths. Z. Ryjáček and P. Vrána in [J. Graph Theory, 66 (2011) 152-173] prove that a fascinating conjecture of Thomassen on hamiltonian line graphs is equivalent to that every essentially 4-edge-connected graph has a 2-spanning-connected line graph. We prove that for any essentially 3-edge-connected graph G and any positive integer s, if G satisfies an Ore degree condition lower bounded by an arbitrary linear function in the number of vertices, then L(G) is s-spanning-connected with only finitely many contraction obstacles. When s=3, we determine a finite graph family J′(n) such that for every simple graph G on n≥156 vertices with κ(L(G))≥3 and satisfying d(u)+d(v)≥2(n−6)5 for any pair of nonadjacent vertices u and v, we have either κ⁎(L(G))≥3 or G is contractible to a member in J′(n).

Decomposing hypergraphs into cycle factors

A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum (k−1)-degree δk−1(H)⩾(1/2+o(1))n, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on n vertices with δ(G)⩾(1/2+o(1))n contains (1−o(1))r edge-disjoint Hamilton cycles where r is the largest integer such that G contains a spanning 2r-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with δk−1(H)⩾(1/2+o(1))n contains (1−o(1))r edge-disjoint (tight) Hamilton cycles where r is the largest integer such that H contains a spanning subgraph with each vertex belonging to kr edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus.In fact, our main result applies to approximately vertex-regular k-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.

Properly colored 2-factors of edge-colored complete bipartite graphs

Let Gc be an edge-colored graph. The minimum color degree of Gc, denoted δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident with v. A cycle in Gc is called properly colored if any of its adjacent edges have distinct colors. A properly colored 2-factor of Gc is a spanning subgraph consisting of vertex-disjoint properly colored cycles of Gc. It was conjectured in the literature that, if δc(Km,n)≥m+n4+1, then each vertex of Km,nc is contained in a properly colored cycle of length l for any even integer l with 4≤l≤min⁡{2m,2n}. In this paper, we consider the existence of properly colored 2-factor in edge-colored bipartite graphs and obtain the following two results: (i) if δc(Kn,n)>3n/4, then Kn,nc contains a properly colored 2-factor. (ii) For every number ε with 0<ε<1/2, there exists an integer n0(ε) such that every Kn,nc with n≥n0(ε) and δc(Kn,n)≥(1/2+ε)n contains a properly colored 2-factor.

On the Sizes of k-edge-maximal r-uniform Hypergraphs

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$. A hypergraph $H'=(V',E')$ is called a subhypergraph of $H=(V,E)$ if $V'\subseteq V$ and $E'\subseteq E$. A $r$-uniform hypergraph $H=(V,E)$ is $k$-edge-maximal if every subhypergraph of $H$ has edge-connectivity at most $k$, but for any edge $e\in E(K_n^r)\setminus E(H)$, $H+e$ contains at least one subhypergraph with edge-connectivity at least $k+1$. Let $k$ and $r$ be integers with $k\geq2$ and $r\geq2$, and let $t=t(k,r)$ be the largest integer such that $(^{t-1}_{r-1})\leq k$. That is, $t$ is the integer satisfies $(^{t-1}_{r-1})\leq k<(^{t}_{r-1})$. We prove that if $H$ is a $r$-uniform $k$-edge-maximal hypergraph such that $n=|V(H)|\geq t$, then ($i$) $|E(H)|\leq (^{t}_{r})+(n-t)k$, and this bound is best possible; ($ii$) $|E(H)|\geq (n-1)k -((t-1)k-(^{t}_{r}))\lfloor\frac{n}{t}\rfloor$, and this bound is best possible. This extends former results in [8] and [6].

Quantization for Spectral Super-Resolution

We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, we define the over-sampling ratio \(\lambda \) as the largest integer such that \(\lfloor M/\lambda \rfloor - 1\ge 4/\Delta \), where M denotes the number of Fourier measurements and \(\Delta \) is the minimum separation distance associated with the atomic measure to be resolved. We prove that for any number \(K\ge 2\) of quantization levels available for the real and imaginary parts of the measurements, our quantization method combined with either TV-min/BLASSO or ESPRIT guarantees reconstruction accuracy of order \(O(M^{1/4}\lambda ^{5/4} K^{- \lambda /2})\) and \(O(M^{3/2} \lambda ^{1/2} K^{- \lambda })\), respectively, where the implicit constants are independent of M, K and \(\lambda \). In contrast, naive rounding or memoryless scalar quantization for the same alphabet offers a guarantee of order \(O(M^{-1}K^{-1})\) only, regardless of the reconstruction algorithm.