E2. Chromatic Number of the Plane (My Favorite Open Problem)

2011 ◽  
pp. 145-150
Author(s):  
Alexander Soifer
Keyword(s):  
Chromatic Number ◽  
Open Problem

scholarly journals On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Christopher Duffy ◽  
Sonja Linghui Shan
Keyword(s):  
Chromatic Number ◽  
Open Problem ◽  
Planar Graphs ◽  
Oriented Graphs ◽  
Tree Width ◽  
Np Complete

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.

scholarly journals Separation dimension and degree

2019 ◽  
pp. 1-10
Author(s):  
ALEX SCOTT ◽  
DAVID R. WOOD
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Open Problem ◽  
Maximum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Axis Parallel ◽  
Parallel Hyperplane ◽  
Separation Dimension

Abstract The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝ d , such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

Packing Chromatic Number of Cycle Related Graphs

2014 ◽  
Vol 4 (1) ◽  
pp. 27
Author(s):  
Albert William ◽  
Roy Santiago ◽  
Indra Rajasingh
Keyword(s):  
Chromatic Number ◽  
Packing Chromatic Number

Packing coloring on subdivision-vertex and subdivision-edge join of cycle Cm with path Pn

2020 ◽  
pp. 627-634
Author(s):  
K. Rajalakshmi ◽  
M. Venkatachalam ◽  
M. Barani ◽  
D. Dafik
Keyword(s):  
Chromatic Number ◽  
Path Graph ◽  
Packing Chromatic Number ◽  
Cycle Graph

The packing chromatic number $\chi_\rho$ of a graph $G$ is the smallest integer $k$ for which there exists a mapping $\pi$ from $V(G)$ to $\{1,2,...,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. In this paper, the authors find the packing chromatic number of subdivision vertex join of cycle graph with path graph and subdivision edge join of cycle graph with path graph.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals On metric chromatic number of comb product of ladder graph

2021 ◽  
Vol 1836 (1) ◽  
pp. 012026
Author(s):  
M Y Rohmatulloh ◽  
Slamin ◽  
A I Kristiana ◽  
Dafik ◽  
R Alfarisi
Keyword(s):  
Chromatic Number ◽  
Ladder Graph
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