On 2-colored graphs and partitions of boxes

10/21/2018
by   Ron Holzman, et al.
0

We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k-1) vertices. This confirms a conjecture of Bucic, Lidicky, Long, and Wagner (arXiv:1805.11278[math.CO]) and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/28/2018

Isolation of k-cliques

For any positive integer k and any n-vertex graph G, let ι(G,k) denote t...
research
11/14/2022

Gallai's Path Decomposition for 2-degenerate Graphs

Gallai's path decomposition conjecture states that if G is a connected g...
research
05/22/2023

Testing Isomorphism of Graphs in Polynomial Time

Given a graph G, the graph [G] obtained by adding, for each pair of vert...
research
08/07/2023

Testing Graph Properties with the Container Method

We establish nearly optimal sample complexity bounds for testing the ρ-c...
research
11/01/2017

Majority Model on Random Regular Graphs

Consider a graph G=(V,E) and an initial random coloring where each verte...
research
07/27/2020

Unfolding cubes: nets, packings, partitions, chords

We show that every ridge unfolding of an n-cube is without self-overlap,...
research
11/24/2017

(Biased) Majority Rule Cellular Automata

Consider a graph G=(V,E) and a random initial vertex-coloring, where eac...

Please sign up or login with your details

Forgot password? Click here to reset