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.

A Note on Primitive Graphs

1972 ◽  
Vol 15 (3) ◽  
pp. 437-440 ◽  
Author(s):  
I. Z. Bouwer ◽  
G. F. LeBlanc
Keyword(s):  
Connected Graph ◽  
Common Vertex ◽  
Connected Subgraph ◽  
Property A ◽  
Vertex Set

Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

scholarly journals DELAUNAY AND DIAMOND TRIANGULATIONS CONTAIN SPANNERS OF BOUNDED DEGREE

2009 ◽  
Vol 19 (02) ◽  
pp. 119-140 ◽  
Author(s):  
PROSENJIT BOSE ◽  
MICHIEL SMID ◽  
DAMING XU
Keyword(s):  
Unit Disk ◽  
Delaunay Triangulation ◽  
Maximum Degree ◽  
Time Algorithm ◽  
Connected Subgraph ◽  
Unit Disk Graph ◽  
Bounded Degree ◽  
Vertex Set ◽  
Range Of Values ◽  
Degree Bound

Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected subgraph G' of G with vertex set V whose maximum degree is at most 14 + ⌈2π/γ⌉. If G is the Delaunay triangulation of V, and γ = 2π/3, we show that G' is a t-spanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is a triangulation satisfying the diamond property, then for a specific range of values of γ dependent on the angle of the diamonds, we show that G' is a t-spanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on γ. If G is the graph consisting of all Delaunay edges of length at most 1, and γ = π/3, we show that a modified version of the algorithm produces a plane subgraph G' of the unit-disk graph which is a t-spanner (for some constant t) of the unit-disk graph of V, whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25.

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 On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Olivier Baudon ◽  
Julien Bensmail ◽  
Rafał Kalinowski ◽  
Antoni Marczyk ◽  
Jakub Przybyło ◽  
...  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Vertex Set ◽  
International Audience ◽  
Positive Integers

Graph Theory International audience A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ &#x007b;1,\textellipsis,k&#x007d;, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.

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 Existentially Closed BIBD Block-Intersection Graphs

10.37236/988 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Neil A. McKay ◽  
David A. Pike
Keyword(s):  
Block Size ◽  
Combinatorial Design ◽  
Block Designs ◽  
Intersection Graphs ◽  
Triple Systems ◽  
Empty Intersection ◽  
Existentially Closed ◽  
Vertex Set ◽  
Block Intersection ◽  
Balanced Incomplete Block

A graph $G$ with vertex set $V$ is said to be $n$-existentially closed if, for every $S \subset V$ with $|S|=n$ and every $T \subseteq S$, there exists a vertex $x \in V-S$ such that $x$ is adjacent to each vertex of $T$ but is adjacent to no vertex of $S-T$. Given a combinatorial design ${\cal D}$ with block set ${\cal B}$, its block-intersection graph $G_{{\cal D}}$ is the graph having vertex set ${\cal B}$ such that two vertices $b_1$ and $b_2$ are adjacent if and only if $b_1$ and $b_2$ have non-empty intersection. In this paper we study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are $n$-existentially closed. We characterise the BIBDs with block size $k \geq 3$ and index $\lambda=1$ that have 2-e.c. block-intersection graphs and establish bounds on the parameters of BIBDs with index $\lambda=1$ that are $n$-e.c. where $n \geq 3$. For $\lambda \geq 2$ and $n \geq 2$, we prove that only simple $\lambda$-fold designs can have $n$-e.c. block-intersection graphs. In the case of $\lambda$-fold triple systems we show that $n \geq 3$ is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with $k=3$ and $\lambda=2$) have 2-e.c. block-intersection graphs.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

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