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Results tagged with graph-theory
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user 43266
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
2
votes
The max-clique chromatic number of a graph
Fedor Petrov, in a comment on my previous answer, asked whether there is a countable graph in which every maximal clique is infinite, and the hypergraph of maximal cliques has infinite chromatic numbe …
- 13.4k
2
votes
Accepted
The max-clique chromatic number of a graph
The answer is yes if infinite graphs are allowed.
Theorem. For any integer $n\ge3$ there is an infinite graph
$G=(V,E)$ such that $\chi_m(G)=\chi(G)=\aleph_0$, and every maximal clique of $G$ has card …
- 13.4k
6
votes
Accepted
Clique and chromatic number when removing an edge
Let $H$ be a graph with $\omega(H)=2$ and $\chi(H)=4$, say the Grötzsch graph. Let $G$ be the graph obtained by taking the disjoint union $H\cup K_2\cup K_4$ and adding edges joining both vertices in …
- 13.4k
3
votes
Accepted
"Spanning trees" for connected linear hypergraphs
Counterexample. Let $\mathbb N=\{1,2,3,\dots\}$. For $n\in\mathbb N$ let $[n]=\{1,2,\dots,n\}$. Let $V=\mathbb N\times\mathbb N$. For $n\in\mathbb N$ let $e_n=\{n\}\times\mathbb N$ and $f_n=[n]\times\ …
- 13.4k
3
votes
Asymptotics for Ramsey Theory
First, the short answer to the general question is that $\frac{A(n,d,k)}{\binom nk}$ is (nonstrictly) increasing in $n$, whence for $n\ge m$ we have
$$A(n,d,k)\ge\frac{A(m,d,k)}{\binom mk}\binom nk\ge …
3
votes
Graph on $\mathbb{N}$ where almost every vertex is shy
Take any locally finite countable graph with infinitely many shy vertices, e.g., the disjoint union of $\aleph_0$ copies of $K_{1,2}$. Identify the vertex set with $\mathbb N$ in such a way that the s …
- 13.4k
4
votes
Accepted
Chromatic number of triangle-free graph $[[n]]^2$ with edges of form $a<b, b<c$
No. $\chi(T_k)\ge n$ iff $k\gt2^{n-1}$; see this Q&A. So the smallest $4$-chromatic graph of the form $T_k$ is $T_9$ which has $36$ vertices, or $35$ if we don't count the isolated vertex; but the sma …
- 13.4k
7
votes
Accepted
Prove or disprove: $R^{n+1} \supseteq R \cap R^2 \cap \cdots \cap R^n$ for every binary rela...
Let $k_n$ be the least integer $k$ such that, for any digraph $D$ of order $n$ and any vertices $x,y\in D$, if there are $x$-$y$ walks of length $1,\dots,k$, then there are $x$-$y$ walks of all positi …
- 13.4k
21
votes
Accepted
Parity and the Axiom of Choice
The Parity Principle follows from the axiom $\mathbf C_2$ (defined below) which is weaker than the Axiom of Choice. I don't know whether the Parity Principle implies $\mathbf C_2$, but that's another …
- 13.4k
8
votes
Seymour's second neighborhood conjecture for infinite graphs
A "digraph" is a simple digraph without $2$-cycles, i.e., an oriented graph. "Locally finite" means outwards locally finite, i.e., each vertex $x$ has finite outdegree $\deg^+(x)\lt\infty$. If $x,y$ a …
- 13.4k
1
vote
Conflict-free coloring of $\mathbb{R}$ with the Euclidean topology
The fact that $\chi_\text{cf}(\mathbb R,\tau)=\aleph_0$ can be generalized as follows. Given a hypergraph $H=(V,E)$ let's say that a set $S\subseteq V$ is a dense set (or a vertex cover) if $S\cap e\n …
- 13.4k
4
votes
Subgraph avoiding colorings
Yes, $P_H(G,t)$ is just the chromatic polynomial of the hypergraph whose vertices are the vertices of $G$ and whose edges are the vertex sets of subgraphs of $G$ that are isomorphic to $H$.
The fact t …
- 13.4k
1
vote
Connected vs strongly connected graphs
Trivially:
A digraph $D$ is strongly connected if and only if it satisfies the two conditions:
(i) the underlying graph of $D$ is connected;
(ii) every arc in $D$ is part of a directed cycle.
If a fin …
- 13.4k
9
votes
The chromatic number of the union of two graphs
This is a comment, not an answer, but it will be more convenient to post it as an answer. Consider the vertices of $G_n$ as subsets of $[n]=\{1,\dots,n\}$.
Observation 1. $\chi(G_n)\ge\left\lfloor\fra …
- 13.4k
6
votes
Induced subgraphs of any given smaller chromatic number
As noted in a comment by Robert Furber, the original question about induced subgraphs is answered by the following simple counterexample due to F. Galvin, Chromatic Numbers of Subgraphs, Periodica Mat …
- 13.4k