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 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.

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 Colouring Planar Mixed Hypergraphs

10.37236/1538 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
André Kündgen ◽  
Eric Mendelsohn ◽  
Vitaly Voloshin
Keyword(s):  
Planar Graph ◽  
Sharp Estimate ◽  
Chromatic Polynomial ◽  
Vertex Set ◽  
Chromatic Spectrum ◽  
Special Case ◽  
Mixed Hypergraph ◽  
Families Of Subsets

A mixed hypergraph is a triple ${\cal H} = (V,{\cal C}, {\cal D})\;$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, the ${\cal C}$-edges and ${\cal D}$-edges, respectively. A $k$-colouring of ${\cal H}$ is a mapping $c: V\rightarrow [k]$ such that each ${\cal C}$-edge has at least two vertices with a ${\cal C}$ommon colour and each ${\cal D}$-edge has at least two vertices of ${\cal D}$ifferent colours. ${\cal H}$ is called a planar mixed hypergraph if its bipartite representation is a planar graph. Classic graphs are the special case of mixed hypergraphs when ${\cal C}=\emptyset$ and all the ${\cal D}$-edges have size 2, whereas in a bi-hypergraph ${\cal C} = {\cal D}$. We investigate the colouring properties of planar mixed hypergraphs. Specifically, we show that maximal planar bi-hypergraphs are 2-colourable, find formulas for their chromatic polynomial and chromatic spectrum in terms of 2-factors in the dual, prove that their chromatic spectrum is gap-free and provide a sharp estimate on the maximum number of colours in a colouring.

scholarly journals Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 828
Author(s):  
Nicholas Newman ◽  
Vitaly Voloshin
Keyword(s):  
Block Designs ◽  
Connected Subgraph ◽  
Vertex Set ◽  
Mixed Hypergraph

In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph H that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle. We propose an algorithm to color the (r,r)-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show χ(H)=2 and χ¯(H)=n−s or n−s−1 where s is the sieve number.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals PELABELAN TOTAL TITIK AJAIB PADA GRAF PETERSEN

2017 ◽  
Vol 1 (1) ◽  
pp. 44
Author(s):  
Chusnul Noeriansyah Poetri
Keyword(s):  
Mapping Functions ◽  
Vertex Set ◽  
Positive Integers

Suppose a graph G with vertex set V(G) and the edge set E(G) where each vertex V(G) and edge E(G) is given a one - one function and on the mapping functions using positive integers {1,2, … ,

scholarly journals On stability and stationary points in nonlinear optimization

1986 ◽  
Vol 28 (1) ◽  
pp. 36-56 ◽  
Author(s):  
J. Guddat ◽  
H. Th. Jongen ◽  
J. Rueckmann
Keyword(s):  
Stability Condition ◽  
Minimization Problem ◽  
Level Sets ◽  
Constraint Qualification ◽  
Stationary Points ◽  
Feasible Set ◽  
Manifold With Boundary ◽  
Feasible Sets ◽  
Constrained Minimization Problem ◽  
Image Position

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).

scholarly journals SINGLE-SOURCE DILATION-BOUNDED MINIMUM SPANNING TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 159-170
Author(s):  
OTFRIED CHEONG ◽  
CHANGRYEOL LEE
Keyword(s):  
Approximation Algorithm ◽  
Spanning Trees ◽  
Edge Length ◽  
Minimum Spanning Trees ◽  
Np Hard ◽  
Single Source ◽  
Vertex Set ◽  
Dilation Factor

Given a set S of points in the plane, a geometric network for S is a graph G with vertex set S and straight edges. We consider a broadcasting situation, where one point r ∊ S is a designated source. Given a dilation factor δ, we ask for a geometric network G such that for every point v ∊ S there is a path from r to v in G of length at most δ|rv|, and such that the total edge length is minimized. We show that finding such a network of minimum total edge length is NP-hard, and give an approximation algorithm.

scholarly journals Paths of specified length in random k-partite graphs

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
C.R. Subramanian
Keyword(s):  
Random Graphs ◽  
Expected Number ◽  
Vertex Set ◽  
International Audience ◽  
Simple Paths ◽  
Positive Integers

International audience Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.

scholarly journals On the Total Vertex Irregular Labeling of Proper Interval Graphs

2020 ◽  
Vol 12 (4) ◽  
pp. 537-543
Author(s):  
A. Rana
Keyword(s):  
Efficient Algorithm ◽  
Simple Graph ◽  
Interval Graphs ◽  
Vertex Weight ◽  
Vertex Set ◽  
Positive Integers ◽  
Irregularity Strength

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers).  For a simple graph G = (V, E) with vertex set V and edge set E, a labeling  Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling  Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which  a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

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