Largest Integer Research Articles

Codegree Thresholds for Covering 3-Uniform Hypergraphs

Given two 3-uniform hypergraphs F and G, we say that G has an F-covering if we can cover V(G) by copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V(G) is contained in at least d triples from E(G). Define c_2(n,F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. This is a natural problem intermediate (but distinct) from the well-studied Tur\'an problems and tiling problems. In this paper, we determine c_2(n, K_4) (for n>98) and the associated extremal configurations (for n>998), where K_4 denotes the complete 3-graph on 4 vertices. We also obtain bounds on c_2(n,F) which are apart by at most 2 in the cases where F is K_4^- (K_4 with one edge removed), K_5^-, and the tight cycle C_5 on 5 vertices.

On the Achromatic Number of Honeycomb Networks

The achromatic number for a graph G= (V, E) is the largest integer m such that there is a partition of Vinto disjoint independent sets (V1, V2, …, Vm) satisfying the condition that for each pair of distinct sets Vi, Vj, Vi∪Vj is not an independent set in G. In this paper we present approximation algorithms to determine the achromatic number for honeycomb networks.

Rainbow cycles in edge-colored graphs

Let G be a graph of order n with an edge coloring c, and let δc(G) denote the minimum color degree of G, i.e., the largest integer such that each vertex of G is incident with at least δc(G) edges having pairwise distinct colors. A subgraph F⊂G is rainbow if all edges of F have pairwise distinct colors. In this paper, we prove that (i) if G is triangle-free and δc(G)>n3+1, then G contains a rainbow C4, and (ii) if δc(G)>n2+2, then G contains a rainbow cycle of length at least 4.

On the b-coloring of tight graphs

A coloring c of a graph $$G=(V,E)$$G=(V,E) is a b-coloring if for every color i there is a vertex, say w(i), of color i whose neighborhood intersects every other color class. The vertex w(i) is called a b-dominating vertex of color i. The b-chromatic number of a graph G, denoted by b(G), is the largest integer k such that G admits a b-coloring with k colors. Let m(G) be the largest integer m such that G has at least m vertices of degree at least $$m-1$$m-1. A graph G is tight if it has exactly m(G) vertices of degree $$m(G)-1$$m(G)-1, and any other vertex has degree at most $$m(G)-2$$m(G)-2. In this paper, we show that the b-chromatic number of tight graphs with girth at least 8 is at least $$m(G)-1$$m(G)-1 and characterize the graphs G such that $$b(G)=m(G)$$b(G)=m(G). Lin and Chang (2013) conjectured that the b-chromatic number of any graph in $$\mathcal {B}_{m}$$Bm is m or $$m-1$$m-1 where $$\mathcal {B}_{m}$$Bm is the class of tight bipartite graphs $$(D,D{^\prime })$$(D,Dź) of girth 6 such that D is the set of vertices of degree $$m-1$$m-1. We verify the conjecture of Lin and Chang for some subclass of $$\mathcal {B}_{m}$$Bm, and we give a lower bound for any graph in $$\mathcal {B}_{m}$$Bm.

Supereulerian graphs with width [formula omitted] and [formula omitted]-collapsible graphs

For an integer s>0 and for u,v∈V(G) with u≠v, an (s;u,v)-trail-system of G is a subgraph H consisting of s edge-disjoint (u,v)-trails. A graph is supereulerian with widths if for any u,v∈V(G) with u≠v, G has a spanning (s;u,v)-trail-system. The supereulerian widthμ′(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer k with 0≤k≤s. Thus a graph G with μ′(G)≥2 has a spanning Eulerian subgraph. Catlin (1988) introduced collapsible graphs to study graphs with spanning Eulerian subgraphs, and showed that every collapsible graph G satisfies μ′(G)≥2 (Catlin, 1988; Lai et al., 2009). Graphs G with μ′(G)≥2 have also been investigated by Luo et al. (2006) as Eulerian-connected graphs. In this paper, we extend collapsible graphs to s-collapsible graphs and develop a new related reduction method to study μ′(G) for a graph G. In particular, we prove that K3,3 is the smallest 3-edge-connected graph with μ′<3. These results and the reduction method will be applied to determine a best possible degree condition for graphs with supereulerian width at least 3, which extends former results in Catlin (1988) and Lai (1988).

On quasi-monotonous graphs

A dominating coloring by k colors is a proper k coloring where every color i has a representative vertex xi adjacent to at least one vertex in each of the other classes. The b-chromatic number, b(G), of a graph G is the largest integer k such that G admits a dominating coloring by k colors. A graph G=(V,E) is said b-monotonous if b(H1)≥b(H2) for every induced subgraph H1 of G and every subgraph H2 of H1. Here we say that a graph G is quasi b-monotonous, or simply quasi-monotonous, if for every vertex v∈V, b(G−v)≤b(G)+1. We shall study the quasi-monotonicity of several classes. We show in particular that chordal graphs are not quasi-monotonous in general, whereas chordal graphs with large b-chromatic number, and (P,coP,chair,diamond)-free graphs are quasi-monotonous. The (P5,P,dart)-free graphs are monotonous. Finally we give new bounds for the b-chromatic number of any vertex deleted subgraph of a chordal graph.

A Twelve Vertex Theorem for 3-Connected Claw-Free Graphs

The cyclability of a graph H, denoted by C(H), is the largest integer r such that H has a cycle through any r vertices. For a claw-free graph H, by Ryjaă?ek (J Comb Theory Ser B 70:217---224, 1997) closure concept, there is a $$K_3$$K3-free graph G such that the closure $$cl(H)=L(G)$$cl(H)=L(G). In this note, we prove that for a 3-connected claw-free graph H with its closure $$cl(H)=L(G)$$cl(H)=L(G), $$C(H)\ge 12$$C(H)?12 if and only if G can not be contracted to the Petersen graph in such a way that each vertex in P is obtained by contracting a nontrivial connected $$K_3$$K3-free subgraph. This is an improvement of the main result in Gyori and Plummer (Stud Sci Math Hung 38:233---244, 2001).

$b$-Chromatic number of Some Splitting Graphs

A $b$-colouring of a graph $G$ is a proper vertex colouring of $G$ such that each colour class contains a vertex that has atleast one neighbour in every other colour class and $b$-chromatic number of a graph $G$ is the largest integer $\phi(G)$ for which $G$ has a $b$-colouring with $\phi(G)$ colours. In this paper, we have obtained the $b$-chromatic number of the splitting graphs of path $P_n$, cycle $C_n$, star $K_{1,n}$, fan graph $F_n$, triangular snake $T_n$, the $H$-graph $H_n$, the corona graph $P_n\circ K_1$ and $C_n\circ K_1$.

The Parametric Frobenius Problem

Given relatively prime positive integers $a_1,\ldots,a_n$, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the $a_i$. We examine the parametric version of this problem: given $a_i=a_i(t)$ as functions of $t$, compute the Frobenius number as a function of $t$. A function $f:\mathbb{Z}_+\rightarrow\mathbb{Z}$ is a quasi-polynomial if there exists a period $m$ and polynomials $f_0,\ldots,f_{m-1}$ such that $f(t)=f_{t\bmod m}(t)$ for all $t$. We conjecture that, if the $a_i(t)$ are polynomials (or quasi-polynomials) in $t$, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large $t$. We prove this in the case where the $a_i(t)$ are linear functions, and also prove it in the case where $n$ (the number of generators) is at most 3.

Perfect competition, methodologically contemplated

:The concept of perfect competition embodies a formal contradiction, precisely as would that of “the largest integer.” The ascription of legitimate meaning to the concept, as in George J. Stigler’s well-known 1957 essay, “Perfect Competition, Historically Contemplated,” is demonstrably circular, hence methodologically unacceptable. Mathematical problems arising from the concept are explored and it is these that give rise to the contradiction, as is demonstrated. Paradoxically, the elimination of perfect competition and its supporting canon comes at no cost to a scientific, that is, empirically oriented economics.

The b-chromatic index of direct product of graphs

The b-chromatic index φ′(G) of a graph G is the largest integer k such that G admits a proper k-edge coloring in which every color class contains at least one edge incident to edges in every other color class. We give in this work bounds for the b-chromatic index of the direct product of graphs and provide general results for many direct products of regular graphs. In addition, we introduce an integer linear programming model for the b-edge coloring problem, which we use for computing exact results for the direct product of some special graph classes.

Disjoint induced subgraphs of the same order and size

For a graph G, let f(G) be the largest integer k such that there are two vertex-disjoint subgraphs of G each on k vertices, both inducing the same number of edges. We prove that f(G)≥n/2−o(n) for every graph G on n vertices. This answers a question of Caro and Yuster.

On the [formula omitted]-coloring of [formula omitted

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbor in all other color classes. In this paper, we show that for every edge e of a graph G, we have b(G−e)≤b(G)+⌈n2⌉−2. Also, we characterize all graphs G for which b(G−e)=b(G)+⌈n2⌉−2 for some edge e of G.

Graphs of girth at least 7 have high b-chromatic number

A b-coloring of a graph is a proper coloring of its vertices such that every color class contains a vertex that has neighbors in all other color classes. The b-chromatic number of a graph is the largest integer b(G) such that the graph has a b-coloring with b(G) colors. This metric is upper bounded by the largest integer m(G) for which G has at least m(G) vertices with degree at least m(G)−1. There are a number of results reporting that graphs with high girth have high b-chromatic number when compared to m(G). Here, we prove that every graph with girth at least 7 has b-chromatic number at least m(G)−1. Our proof also yields a polynomial algorithm that produces an optimal b-coloring of these graphs.

Frobenius numbers of Pythagorean triples

Given relatively prime integers a1,…, an, the Frobenius number g(a1,…, an) is defined as the largest integer which cannot be expressed as x1a1 +⋯+ xnan with xi nonnegative integers. In this paper, we give the Frobenius number of primitive Pythagorean triples:

An improved estimate for the condition number anomaly of univariate Gaussian correlation matrices

In this short note, it is proved that the derivatives of the parametrized univariate Gaussian correlation matrix R_g (θ) = (exp(−θ(x_i − x_j )^2_{i,j} ∈ R^{n×n} are rank-deficient in the limit θ = 0 up to any order m &lt; (n − 1)/2. This result generalizes the rank deficiency theorem for Euclidean distance matrices, which appear as the first-order derivatives of the Gaussian correlation matrices in the limit θ = 0. As a consequence, it is shown that the condition number of R_g(θ) grows at least as fast as 1(/θ^(mˆ +1) for θ → 0, where mˆ is the largest integer such that mˆ &lt; (n − 1)/2. This considerably improves the previously known growth rate estimate of 1/θ^22 for the so-called Gaussian condition number anomaly.

On a conjecture by Wilf about the Frobenius number

Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$.

An upper bound on the double domination number of trees

In a graph G, a vertex dominates itself and its neighbors. A set S of vertices in a graph G is a double dominating set if S dominates every vertex of G at least twice. The double domination number γx2(G) is the minimum cardinality of a double dominating set in G. 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 edges in G. In this paper, we show that for any tree T of order n ≥ 2, different from P4, γx2(T) ≤ 3a(T)+1. 2.

On b-vertex and b-edge critical graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b + -vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In this note, we explain some properties in b + -vertex

Distinct Volume Subsets

Suppose that $a$ and $d$ are positive integers with $a \geq 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in $\mathbb{R}^d$ contains a subset of $t$ points for which all the nonzero volumes of the ${t \choose a}$ subsets of order $a$ are distinct. Beginning with Erdos in 1957, the function $h_{2,d}(n)$ has been closely studied and is known to be at least a power of $n$. We improve the best known bound for $h_{2,d}(n)$ and show that $h_{a,d}(n)$ is at least a power of $n$ for all $a$ and $d$.