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Results tagged with graph-theory
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user 8628
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.
5
votes
3
answers
253
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The max-clique chromatic number of a graph
Let $G = (V,E)$ be a graph. Every clique, that is, complete subgraph, is
contained in a maximal clique with respect to $\subseteq$ (this is
an easy consequence of Zorn's Lemma). Let $\newcommand{\MC}{ …
- 48.9k
1
vote
1
answer
46
views
"Gray code" for $[\omega]^{<\omega}$
Let $\newcommand{\oo}{[\omega]^{<\omega}}\oo$ denote the collection of finite subsets of the set of non-negative integers $\newcommand{\o}{\omega}\o$.
If $A,B$ are any sets, let $A \,\triangle \, B = …
- 48.9k
0
votes
1
answer
94
views
Countable graph with $2^{\aleph_0}$ non-isomorphic induced minors
Let $G=(V,E)$ be a simple, undirected graph. If $S, T\subseteq V$ are disjoint sets, we say that $S,T$ are connected to each other if there are $s\in S, t\in T$ such that $\{s,t\}\in E$. We say a grap …
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0
votes
1
answer
69
views
Forced monochromatic pairs in graphs
Starting point. Consider the "$V$-graph" on the vertex set $\{1,2,3\}$ and let the edges be $\{1,2\}$ and $\{2,3\}$. This graph is clearly bipartite. It is a trivial observation that whenever we color …
- 48.9k
5
votes
1
answer
397
views
4-color theorem for hypergraphs
Question. Does every hypergraph that does not admit a complete minor with $5$ elements have a coloring with $4$ colors?
Below are the definitions to make this precise.
If $H = (V, E)$ is a hypergraph …
- 48.9k
0
votes
1
answer
79
views
Infinite complete minor in $\min,\max$-graph on $\mathbb{N}$
Let $[\omega]^2 =\big\{\{x,y\}:x\neq y \in \omega\big\}$ denote the collection of all 2-element subsets of the non-negative integers. Let $$E=\big\{\{p,q\} : p,q \in [\omega]^2 \text{ and } \max(p)=\m …
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3
votes
1
answer
233
views
Clique and chromatic number when removing an edge
For any set $X$, let $[X]^2=\big\{\{x,y\}: x\neq y\in X\big\}$. If $G=(V,E)$ is a simple undirected graph and $e\in E$, let $G\setminus e = \big(V\setminus e, E \cap [V\setminus e]^2\big)$.
If $G=(V,E …
- 48.9k
0
votes
1
answer
61
views
Is every connected edge-swapping graph edge-transitive?
We say that a connected, simple, undirected graph $G=(V,E)$ is edge-swapping if for every $e\in E$ there is a graph isomorphism $\varphi:G\to G$ such that for the restriction $\varphi|_e$ we have $\te …
- 48.9k
2
votes
1
answer
97
views
"Spanning trees" for connected linear hypergraphs
Starting point. For every simple, undirected graph $G=(V,E)$ there is $E_0\subseteq E$ such that $(V,E_0)$ is minimally connected: the spanning tree. The goal of this question is to find out whether t …
- 48.9k
12
votes
1
answer
678
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Graphs $G$ with $G \cong \text{Aut}(G)$
Let $G=(V,E)$ be a simple, undirected graph. By $\newcommand{\Aut}{\text{Aut}}\Aut(G)$ we denote the collection of graph isomorphisms $\varphi:G\to G$. We let $$E(\Aut(G)) =\big\{\{\varphi, \psi\}:\va …
- 48.9k
7
votes
1
answer
152
views
$|G|/\alpha(G) \leq \eta(G)$ where $\eta(G)$ is the Hadwiger number
Let $G=(V,E)$ be a finite, simple, undirected graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$.
Hadwiger's celebrated conjecture states that $\chi …
- 48.9k
1
vote
1
answer
149
views
Chromatic number of the insert-and-shift graph on $S_n$
Let $S_n$ be the collection of bijections $\varphi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. In an earlier question, the insert-and-shift graph structure was introduced on $S_n$ and the resulting graph is ca …
- 48.9k
4
votes
0
answers
228
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Optimal colorings
If $G=(V,E)$ is a graph, we call the smallest cardinal $\kappa$ such that there is a coloring map $c:V\to \kappa$ as the chromatic number of $G$ and denote it by $\chi(G)$.
For any coloring $c:V(G) \t …
- 48.9k
1
vote
3
answers
184
views
Graph on $\mathbb{N}$ where almost every vertex is shy
The following question is loosely based on the friendship paradox.
Let $G=(V,E)$ be a simple, undirected graph. For $v\in V$, we let the neighborhood of $v$ be $N(v) = \big\{w\in V:\{v,w\}\in E\big\}$ …
- 48.9k
6
votes
1
answer
218
views
Chromatic number of the infinite Erdős–Hajnal shift-graph
For any set $X$, let $[X]^2= \big\{\{x,y\}: x\neq y \in X\big\}$. Let $\kappa$ be an infinite cardinal. Let $G_\kappa = ([\kappa]^2, E_\kappa)$ where $E_\kappa = \big\{\{a,b\}\in \big[[\kappa]^2\big]^ …
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