scholarly journals The Smallest One-Realization of a Given Set

10.37236/1171 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ping Zhao ◽  
Kefeng Diao ◽  
Kaishun Wang
Keyword(s):  
Open Problem ◽  
Feasible Set ◽  
Minimum Number ◽  
Chromatic Spectrum ◽  
Positive Integers ◽  
Mixed Hypergraph

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Král proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we  determine the number  of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král  in 2004.

scholarly journals On Feasible Sets of Mixed Hypergraphs

10.37236/1772 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Daniel Král
Keyword(s):  
Vertex Coloring ◽  
Np Hard ◽  
Finite Sets ◽  
Feasible Set ◽  
Feasible Sets ◽  
Vertex Set ◽  
Positive Integers ◽  
Mixed Hypergraph ◽  
Families Of Subsets

A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, called ${\cal C}$-edges and ${\cal D}$-edges. A vertex coloring of $H$ is proper if each ${\cal C}$-edge contains two vertices with the same color and each ${\cal D}$-edge contains two vertices with different colors. The spectrum of $H$ is a vector $(r_1,\ldots,r_m)$ such that there exist exactly $r_i$ different colorings using exactly $i$ colors, $r_m\ge 1$ and there is no coloring using more than $m$ colors. The feasible set of $H$ is the set of all $i$'s such that $r_i\ne 0$. We construct a mixed hypergraph with $O(\sum_i\log r_i)$ vertices whose spectrum is equal to $(r_1,\ldots,r_m)$ for each vector of non-negative integers with $r_1=0$. We further prove that for any fixed finite sets of positive integers $A_1\subset A_2$ ($1\notin A_2$), it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to $A_2$ even if it is promised that it is either $A_1$ or $A_2$. This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard.

scholarly journals Vertex connectivity of the power graph of a finite cyclic group II

2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Vertex Connectivity ◽  
Induced Subgraph ◽  
Power Graph ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
Positive Integers ◽  
Appl Math

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

scholarly journals On ranges of variants of the divisor functions that are dense

2015 ◽  
Vol 11 (06) ◽  
pp. 1905-1912 ◽  
Author(s):  
Colin Defant
Keyword(s):  
Real Number ◽  
Open Problem ◽  
Dense Subset ◽  
Arithmetic Function ◽  
Divisor Functions ◽  
Positive Integers ◽  
Multiplicative Arithmetic ◽  
Multiplicative Arithmetic Function

For a real number t, let st be the multiplicative arithmetic function defined by [Formula: see text] for all primes p and positive integers α. We show that the range of a function s-r is dense in the interval (0, 1] whenever r ∈ (0, 1]. We then find a constant ηA ≈ 1.9011618 and show that if r > 1, then the range of the function s-r is a dense subset of the interval [Formula: see text] if and only if r ≤ ηA. We end with an open problem.

scholarly journals Strong T-Coloring of Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 4677-4681
Keyword(s):  
Chromatic Number ◽  
Maximum Value ◽  
Finite Set ◽  
Minimum Number ◽  
Edge Span ◽  
Positive Integers

A 𝑻-coloring of a graph 𝑮 = (𝑽,𝑬) is the generalized coloring of a graph. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set T of positive integers containing 𝟎 , a 𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻. We define Strong 𝑻-coloring (S𝑻-coloring , in short), as a generalization of 𝑻-coloring as follows. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set 𝑻 of positive integers containing 𝟎, a S𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻 and |𝒇(𝒖) − 𝒇(𝒘)| ≠ |𝒇(𝒙) − 𝒇(𝒚)| for any two distinct edges 𝒖𝒘, 𝒙𝒚 in 𝑬(𝑮). The S𝑻-Chromatic number of 𝑮 is the minimum number of colors needed for a S𝑻-coloring of 𝑮 and it is denoted by 𝝌𝑺𝑻(𝑮) . For a S𝑻 coloring 𝒄 of a graph 𝑮 we define the 𝒄𝑺𝑻- span 𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all pairs 𝒖, 𝒗 of vertices of 𝑮 and the S𝑻 -span 𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒔𝒑𝑺𝑻(𝑮) = min 𝒔𝒑𝑺𝑻 𝒄 (𝑮) where the minimum is taken over all ST-coloring c of G. Similarly the 𝒄𝑺𝑻-edgespan 𝒆𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all edges 𝒖𝒗 of 𝑮 and the S𝑻-edge span 𝒆𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒆𝒔𝒑𝑺𝑻(𝑮) = min 𝒆𝒔𝒑𝑺𝑻 𝒄 𝑮 where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S𝑻- chromatic number, 𝒔𝒑𝑺𝑻(𝑮) , and 𝒆𝒔𝒑𝑺𝑻(𝑮) of graphs.

scholarly journals On the Upper and Lower Chromatic Numbers of BSQSs(16)

10.37236/1550 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Giovanni Lo Faro ◽  
Lorenzo Milazzo ◽  
Antoinette Tripodi
Keyword(s):  
Chromatic Number ◽  
Chromatic Numbers ◽  
New Information ◽  
Chromatic Spectrum ◽  
Minimum Numbers ◽  
Mixed Hypergraph

A mixed hypergraph is characterized by the fact that it possesses ${\cal C}$-edges as well as ${\cal D}$-edges. In a colouring of a mixed hypergraph, every ${\cal C}$-edge has at least two vertices of the same colour and every ${\cal D}$-edge has at least two vertices coloured differently. The upper and lower chromatic numbers $\bar{\chi}$, $\chi$ are the maximum and minimum numbers of colours for which there exists a colouring using all the colours. The concepts of mixed hypergraph, upper and lower chromatic numbers are applied to $SQSs$. In fact a BSQS is an SQS where all the blocks are at the same time ${\cal C}$-edges and ${\cal D}$-edges. In this paper we prove that any $BSQS(16)$ is colourable with the upper chromatic number $\bar{\chi}=3$ and we give new information about the chromatic spectrum of BSQSs($16$).

Distance Vertex Identification in Graphs

2021 ◽  
pp. 2150005
Author(s):  
Gary Chartrand ◽  
Yuya Kono ◽  
Ping Zhang
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Identification Number ◽  
Minimum Number ◽  
Positive Integers ◽  
Distance Formula

A red-white coloring of a nontrivial connected graph [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and the [Formula: see text]th coordinate of the code is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The problem of determining those graphs that are ID-graphs is investigated. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text] and is denoted by [Formula: see text]. It is shown that (1) a nontrivial connected graph [Formula: see text] has ID-number 1 if and only if [Formula: see text] is a path, (2) the path of order 3 is the only connected graph of diameter 2 that is an ID-graph, and (3) every positive integer [Formula: see text] different from 2 can be realized as the ID-number of some connected graph. The identification spectrum of an ID-graph [Formula: see text] is the set of all positive integers [Formula: see text] such that [Formula: see text] has an ID-coloring with exactly [Formula: see text] red vertices. Identification spectra are determined for paths and cycles.

scholarly journals Filling of closed surfaces

2018 ◽  
Vol 10 (04) ◽  
pp. 897-913 ◽  
Author(s):  
Bidyut Sanki
Keyword(s):  
Disjoint Union ◽  
Mapping Class ◽  
Constructive Proof ◽  
Closed Curves ◽  
Closed Surfaces ◽  
Minimal Filling ◽  
Minimum Number ◽  
Genus Formula ◽  
Positive Integers ◽  
Closed Oriented Surface

Let [Formula: see text] denote a closed oriented surface of genus [Formula: see text]. A set of simple closed curves is called a filling of [Formula: see text] if its complement is a disjoint union of discs. The mapping class group [Formula: see text] of genus [Formula: see text] acts on the set of fillings of [Formula: see text]. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of [Formula: see text] are in the same [Formula: see text]-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of [Formula: see text] whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of [Formula: see text]. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of [Formula: see text] is two. Finally, given positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we construct a filling pair of [Formula: see text] such that the complement is a union of [Formula: see text] topological discs.

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

scholarly journals Thue-like Sequences and Rainbow Arithmetic Progressions

10.37236/1660 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Jaroslaw Grytczuk
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Lovász Local Lemma ◽  
Fixed Integer ◽  
Local Lemma ◽  
Minimum Number ◽  
Lovasz Local Lemma ◽  
Positive Integers

A sequence $u=u_{1}u_{2}...u_{n}$ is said to be nonrepetitive if no two adjacent blocks of $u$ are exactly the same. For instance, the sequence $a{\bf bcbc}ba$ contains a repetition $bcbc$, while $abcacbabcbac$ is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set $\{a,b,c\}$. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this paper we consider a stronger property defined as follows. Let $k\geq 2$ be a fixed integer and let $C$ denote a set of colors (or symbols). A coloring $f:{\bf N}\rightarrow C$ of positive integers is said to be $k$-nonrepetitive if for every $r\geq 1$ each segment of $kr$ consecutive numbers contains a $k$-term rainbow arithmetic progression of difference $r$. In particular, among any $k$ consecutive blocks of the sequence $f=f(1)f(2)f(3)...$ no two are identical. By an application of the Lovász Local Lemma we show that the minimum number of colors in a $k$-nonrepetitive coloring is at most $2^{-1}e^{k(2k-1)/(k-1)^{2}}k^{2}(k-1)+1$. Clearly at least $k+1$ colors are needed but whether $O(k)$ suffices remains open. This and other types of nonrepetitiveness can be studied on other structures like graphs, lattices, Euclidean spaces, etc., as well. Unlike for the classical Thue sequences, in most of these situations non-constructive arguments seem to be unavoidable. A few of a range of open problems appearing in this area are presented at the end of the paper.

scholarly journals On Planar Mixed Hypergraphs

10.37236/1579 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Daniel Král'
Keyword(s):  
Chromatic Number ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vertex Coloring ◽  
Incidence Graph ◽  
Feasible Set ◽  
Vertex Set ◽  
Bridgeless Graph ◽  
Np Complete ◽  
Mixed Hypergraph

A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is its vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, ${\cal C}$–edges and ${\cal D}$–edges. A mixed hypergraph is a bihypergraph iff ${\cal C}={\cal D}$. A hypergraph is planar if its bipartite incidence graph is planar. A vertex coloring of $H$ is proper if each ${\cal C}$–edge contains two vertices with the same color and each ${\cal D}$–edge contains two vertices with different colors. The set of all $k$'s for which there exists a proper coloring using exactly $k$ colors is the feasible set of $H$; the feasible set is called gap-free if it is an interval. The minimum (maximum) number of the feasible set is called a lower (upper) chromatic number. We prove that the feasible set of any planar mixed hypergraph without edges of size two and with an edge of size at least four is gap-free. We further prove that a planar mixed hypergraph with at most two ${\cal D}$–edges of size two is two-colorable. We describe a polynomial-time algorithm to decide whether the lower chromatic number of a planar mixed hypergraph equals two. We prove that it is NP-complete to find the upper chromatic number of a mixed hypergraph even for 3-uniform planar bihypergraphs. In order to prove the latter statement, we prove that it is NP-complete to determine whether a planar 3-regular bridgeless graph contains a $2$-factor with at least a given number of components.

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