Minimum Integer Research Articles

A new lower bound on Hadwiger-Debrunner numbers in the plane

A family of sets \({\cal F}\) is said to satisfy the (p, q) property if among any p sets in \({\cal F}\) some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d + 1 there exists a minimum integer c = HDd(p, q), such that any finite family of convex sets in ℝd that satisfies the (p, q) property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on HDd(p, q), known as ‘the Hadwiger-Debrunner numbers’, is still a major open problem in discrete and computational geometry.The best known upper bound on the Hadwiger-Debrunner numbers in the plane is \(O({p^{(1.5 + \delta)(1 + {1 \over {q - 2}})}})\) (for any δ > 0 and p ≥ q ≥ q0(δ)), obtained by combining the results of Keller, Smorodinsky and Tardos (2017) and of Rubin (2018). The best lower bound is \(H\,{D_2}(p,q) = \Omega \left({{p \over q}\log \left({{p \over q}} \right)} \right)\), obtained by Bukh, Matoušek and Nivasch more than 10 years ago.In this paper we improve the lower bound significantly by showing that HD2(p, q) ≥ p1+Ω(1/q). Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q) problem to an old problem of Erdős on points in general position in the plane. We use a novel construction for Erdős’ problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD2(p, 3). We then generalize the bound to HD2(p, q) for q ≥ 3.

Gallai–Ramsey numbers for graphs with chromatic number three

Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) is defined to be the minimum integer n such that every k-edge coloring of the complete graph Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of H. In this paper, we study Gallai–Ramsey numbers for graphs with chromatic number three such as K̂m for m≥2, where K̂m is a kipas with m+1 vertices obtained from the join of K1 and Pm, and a class of graphs with five vertices, denoted by ℋ. We first study the general lower bound of such graphs and propose a conjecture for the exact value of GRk(K̂m). Then we give a unified proof to determine the Gallai–Ramsey numbers for many graphs in ℋ and obtain the exact value of GRk(K̂4) for k≥1. Our outcomes not only indicate that the conjecture on GRk(K̂m) is true for m≤4, but also imply several results on GRk(H) for some H∈ℋ which are proved individually in different papers.

Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-k-coloring) is an edge-coloring (with colors from {1,2,…,k}) of a complete graph without rainbow triangles. Given a graph H and a positive integer k, the k-colored Gallai-Ramsey number GRk(H) is the minimum integer n such that every Gallai-k-coloring of the complete graph Kn contains a monochromatic copy of H. In this paper, we consider two extremal problems related to Gallai-k-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a k-edge-coloring of Kn. Second, for n≥GRk(K3), we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-k-coloring of Kn, yielding the exact value for k=3. Furthermore, we determine the Gallai-Ramsey number GRk(K4+e) for the graph on five vertices consisting of a K4 with a pendant edge.

On the proper orientation number of chordal graphs

An orientation D of a graph G=(V,E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v∈V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u)≠dD−(v), for all uv∈E(G). An orientation with maximum indegree at most k is called a k-orientation. The proper orientation number of G, denoted by χ→(G), is the minimum integer k such that G admits a proper k-orientation. We prove that determining whether χ→(G)≤k is NP-complete for chordal graphs of bounded diameter, but can be solved in linear-time in the subclass of quasi-threshold graphs. When parameterizing by k, we argue that this problem is FPT for chordal graphs and argue that no polynomial kernel exists, unless NP⊆coNP/poly. We present a better kernel to the subclass of split graphs and a linear kernel to the class of cobipartite graphs.Concerning bounds, we first prove that if G is split, then χ→(G)≤2ω(G)−2 and that if G is a k-uniform block graph, then χ→(G)≤3k−2. These bounds are tight. We also present new families of trees having proper orientation number at most 2 and at most 3. Actually, we prove a general bound stating that any graph G having no adjacent vertices of degree at least c+1 has proper orientation number at most c. This implies new classes of (outer)planar graphs with bounded proper orientation number. We also prove that maximal outerplanar graphs G whose weak-dual is a path satisfy χ→(G)≤13. Finally, we present simple bounds to the classes of chordal claw-free graphs and cographs.

Open -ranks with respect to Segre and Veronese varieties

Let be an integral and non-degenerate variety. Recall (A. Bialynicki-Birula, A. Schinzel, J. Jelisiejew and others) that for any the open rank is the minimal positive integer such that for each closed set there is a set with and , where denotes the linear span. For an arbitrary we give an upper bound for in terms of the upper bound for when is a point in the maximal proper secant variety of and a similar result using only points with submaximal border rank. We study when is a Segre variety (points with -rank and ) and when is a Veronese variety (points with -rank or with border rank ).

An improved approximation algorithm for the minimum common integer partition problem

Given a collection of multisets {X1,X2,…,Xk} (k≥2) of positive integers, a multiset S is a common integer partition for them if S is an integer partition of every multiset Xi,1≤i≤k. The minimum common integer partition (k-MCIP) problem is defined as to find a CIP for {X1,X2,…,Xk} with the minimum cardinality. We present a 65-approximation algorithm for the 2-MCIP problem, improving the previous best algorithm of performance ratio 54 designed by Chen et al. in 2006. We then extend it to obtain an absolute 0.6k-approximation algorithm for k-MCIP when k is even (when k is odd, the approximation ratio is 0.6k+0.4).

Planar graphs without intersecting 5-cycles are signed-4-choosable

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].

Gallai and [formula omitted]-uniform Ramsey numbers of complete bipartite graphs

A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k, the Gallai Ramsey number GRk(H) is the minimum integer N such that every Gallai k-coloring of KN contains a monochromatic copy of H. A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and ℓ, the ℓ-uniform Ramsey number Rkℓ(H) is the minimum integer N such that any uniform k-coloring of any complete multipartite graph on N vertices with each part of cardinality no more than ℓ, contains a monochromatic copy of H. Let Km,n denote a complete bipartite graph. Wu et al. conjectured that GRk(Km,n)=(n−1)(k−2)+R21(Km,n) if R21(Km,n)≥3m−2, where k≥3 and m≥n≥2. In this paper, we show that GRk(Km,n)=(n−1)(k−2)+R2m−1(Km,n) for all integers k≥3 and m≥n≥2. Based on this result, we improve the known general bounds for GRk(Km,n), and confirm the conjecture for some complete bipartite graphs.

Incidence coloring of Mycielskians with fast algorithm

An incidence of a graph G is a vertex-edge pair (v,e) such that the vertex v is incident with the edge e. A proper incidence k-coloring of a graph is a coloring of its incidences involving k colors so that two incidences (u,e) and (w,f) receive distinct colors if and only if u=w, or e=f, or uw∈{e,f}. In this paper, we present some idea of using the incidence coloring to model a kind of multi-frequency assignment problem, in which each transceiver can be simultaneously in both sending and receiving modes, and then establish some theoretical and algorithmic aspects of the incidence coloring.Specifically, we conjecture that if G is the Mycielskian of some graph then it has a proper incidence (Δ(G)+2)-coloring. Actually, our conjecture is motivated by the “(Δ+2) conjecture” of Brualdi and Quinn Massey in 1993, which states that every graph G has a proper incidence (Δ(G)+2)-coloring, and was disproved in 1997 by Guiduli, who pointed out that the Paley graphs with large maximum degree are counterexamples (yet they are all known counterexamples to the “(Δ+2) conjecture”, and are not Mycielskians of any graph).To support our conjecture, we prove in this paper that if G is the Mycielskian of a graph H with |H|≥3Δ(H)+2, then we can construct a proper incidence (Δ(G)+1)-coloring of G in cubic time, and if G is the Mycielskian of an incidence (Δ(H)+1)-colorable graph H with |H|≤2Δ(H), or the Mycielskian of an incidence (Δ(H)+2)-colorable graph H with |H|≥2Δ(H)+1, then G has a proper incidence (Δ(G)+2)-coloring. The minimum positive integer k such that the Mycielskian of a cycle or a path has a proper incidence k-coloring is also determined.

On the size of subsets of $$\mathbb{F}_p^n$$ without p distinct elements summing to zero

Let us fix a prime p. The Erdős-Ginzburg-Ziv problem asks for the minimum integer s such that any collection of s points in the lattice ℤn contains p points whose centroid is also a lattice point in ℤn. For large n, this is essentially equivalent to asking for the maximum size of a subset of \(\mathbb{F}_p^n\) without p distinct elements summing to zero.In this paper, we give a new upper bound for this problem for any fixed prime p ≥ 5 and large n. In particular, we prove that any subset of \(\mathbb{F}_p^n\) without p distinct elements summing to zero has size at most \({C_p} \cdot {\left( {2\sqrt p } \right)^n}\), where Cp is a constant only depending on p. For p and n going to infinity, our bound is of the form p(1/2)·(1+o(1))n, whereas all previously known upper bounds were of the form p(1−o(1))n (with pn being a trivial bound).Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds.

Alfa Laval AB, Sweden

A Gallai coloring of a complete graph is a coloring of the edges without rainbow triangles (all edges colored differently). Given a graph H and a positive integer k, the Gallai Ramsey number GRk(H) is the minimum integer N such that every Gallai k-coloring of KN contains a monochromatic copy of H. A uniform coloring of a complete multipartite graph is a coloring of the edges such that the edges between any two parts receive the same color. Given a graph H and positive integers k and ℓ, the ℓ-uniform Ramsey number Rkℓ(H) is the minimum integer N such that any uniform k-coloring of any complete multipartite graph on N vertices with each part of cardinality no more than ℓ, contains a monochromatic copy of H. Let Km,n denote a complete bipartite graph. Wu et al. conjectured that GRk(Km,n)=(n−1)(k−2)+R21(Km,n) if R21(Km,n)≥3m−2, where k≥3 and m≥n≥2. In this paper, we show that GRk(Km,n)=(n−1)(k−2)+R2m−1(Km,n) for all integers k≥3 and m≥n≥2. Based on this result, we improve the known general bounds for GRk(Km,n), and confirm the conjecture for some complete bipartite graphs.

A patient risk model to determine the optimal output constancy check frequency for a radiotherapy machine.

The output constancy check, a basic quality control (QC) item for radiotherapy machines, is performed daily according to suggestions in technical reports by experienced experts. In this study, a patient risk model was built to determine the optimal frequency of an output constancy check for a specific radiotherapy machine. The method was based on the patient risk model and comprised three steps: 1) the power function graph was used to select a proper QC rule and the average number of QC measurements per QC rule evaluation. 2) The optimal QC frequency was determined by the minimum integer value of expected patients treated between QC measurements. 3) The individual control chart (I-Chart) was used to evaluate the effectiveness of the model. The model was implemented on the output constancy check of a Tomotherapy machine. The QC rule with the limits set to the mean±3 standard deviations and 5 measurements per QC were selected according to the power function graph. The optimal frequency was observed every 21 patients. The I-Chart showed that the optimal frequency detected the machine failure earlier compared to the conventional daily frequency. The model could monitor whether Tomotherapy machine was in good condition and predicted the time to adjust the machine. The optimal output constancy check frequency of a radiotherapy machine is determined by the number of patients, which uses patient risk model. The optimal frequency is superior to the conventional daily frequency in identifying machine failure earlier.

Minimal divergence for border rank-2 tensor approximation

ABSTRACT A tensor is the sum of at least elementary tensors. In addition, a ‘border rank’ is defined: holds if r is the minimum integer such that is a limit of rank-r tensors. Usually, the set of rank-r tensors is not closed, i.e. tensors with may exist. It is easy to see that in such a case the representation of rank-r tensors contains diverging elementary tensors as approaches In a first part, we recall results about the uniform strength of the divergence in the case of general nonclosed tensor formats (restricted to finite dimensions). The second part discusses the r-term format for infinite-dimensional tensor spaces. It is shown that the general situation is very similar to the behaviour of finite-dimensional model spaces. The third part contains the main result: it is proved that in the case of the divergence strength is , i.e. if and the parameters of increase at least proportionally to

On star 5-colorings of sparse graphs

A stark-coloring of a graph G is a proper (vertex) k-coloring of G such that the vertices on a path of length three receive at least three colors. Given a graph G, its star chromatic number, denoted χs(G), is the minimum integer k for which G admits a star k-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the maximum average degree, denoted mad(G), of a graph G is max2|E(H)||V(H)|:H⊂G. It is known that for a graph G, if mad(G)<83, then χs(G)≤6 (Kündgen and Timmons, 2010), and if mad(G)<187 and its girth is at least 6, then χs(G)≤5 (Bu et al., 2009). We improve both results by showing that for a graph G, if mad(G)≤83, then χs(G)≤5. As an immediate corollary, we obtain that a planar graph with girth at least 8 has a star 5-coloring, improving the best known girth condition for a planar graph to have a star 5-coloring (Kündgen and Timmons, 2010; Timmons, 2008).

Optimal and Heuristic Approaches to Modulo Scheduling With Rational Initiation Intervals in Hardware Synthesis

A well-known approach for generating custom hardware with high throughput and low resource usage is modulo scheduling, in which the number of clock cycles between successive inputs (the initiation interval, II) can be lower than the latency of the computation. The II is traditionally an integer, but in this paper, we explore the benefits of allowing it to be a rational number. A rational II can be interpreted as the average number of clock cycles between successive inputs. Since the minimum rational II can be less than the minimum integer II, higher throughput is possible; moreover, allowing rational IIs gives more options in a design-space exploration. We formulate rational-II modulo scheduling as an integer linear programming (ILP) problem that is able to find latency-optimal schedules for a fixed rational II. We also propose two heuristic approaches that make rational-II scheduling more feasible: one based on identifying strongly connected components in the data-flow graph, and one based on iteratively relaxing the target II until a solution is found. We have applied our methods to a standard benchmark of hardware designs, and our results demonstrate an average speedup w.r.t. II of 1.24× in 35% of the encountered scheduling problems compared to state-of-the-art formulations.

On the complexity of coloring ‐graphs

AbstractAn ‐graph is a graph that can be partitioned into r independent sets and ℓ cliques. In the Graph Coloring problem, we are given as input a graph G, and the objective is to determine the minimum integer k such that G admits a proper vertex k‐coloring. In this work, we describe a Poly versus NP‐hard dichotomy of this problem regarding the parameters r and ℓ of ‐graphs, which determine the boundaries of the ‐hardness of Graph Coloring for such classes. We also analyze the complexity of the problem on ()‐graphs under the parameterized complexity perspective. We show that given a (2, 1)‐partition of a graph G, finding an optimal coloring of G is ‐complete even when K1 is a maximal clique of size 3; XP but W[1]‐hard when parameterized by ; fixed‐parameter tractable (FPT) and admits a polynomial kernel when parameterized by . Besides, concerning the case where K1 is a maximal clique of size 3, a P versus NPh dichotomy regarding the neighborhood of K1 is provided; furthermore, an FPT algorithm parameterized by the number of vertices having no neighbor in K1 is presented.

The sum necessary to ensure that a degree sequence pair has an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(A,m,n) for |A|=k and m≥n≥2, where k≥6 is an even integer, and the values of σ(A,m,n) for |A|=4 and m≥n≥3. Therefore, together with some previous results [(Yin, 2016); (Yin and Dai, 2017); (Guan and Yin, 2018)], the values of σ(A,m,n) for |A|≥3 are determined completely.

Tight Gaps in the Cycle Spectrum of 3-Connected Planar Graphs

For any positive integer $k$, define $f(k)$ (respectively, $f_3(k)$) to be the minimal integer $\ge k$ such that every 3-connected planar graph $G$ (respectively, 3-connected cubic planar graph $G$...

Distance-2 Irregular chromatic numbers for some graphs

Let G be a graph and let c:V(G)→{1,2…..k} be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code (v) = (a0, a1, a2….ak) where a0 is the color assigned to v and for 1 ⩽ i ⩽ k, ai is the number of the vertices of G adjacent to that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number of G is the minimum positive integer k for which G has a recognizable k-coloring. In this paper we introduced a new variation of above parameter namely distance-2 irregular coloring. We initiate a study of this parameter and also find the distance 2-irregular chromatic number of some standard graphs.

Lie identities and images of Lie polynomials for the skew-symmetric elements of UTm

ABSTRACT Let UT m be the algebra of all m × m upper triangular matrices over a field whose characteristic is different from 2. Given an involution * of the first kind on UT m , we will obtain the minimal integer t such that the Lie polynomial [[z 1, z 2], …, [z 2t−1, z 2t ]] is an identity for . Afterwards, under a mild technical restriction on , we will describe all multilinear Lie polynomials whose image evaluated on K(m, *) is the space formed by all strictly upper triangular matrices of K(m, *).