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Results tagged with graph-theory
Search options questions only
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user 1737
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.
3
votes
0
answers
177
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Cubic graphs, unfriendly partitions and hamiltonicity
We say that a $2$-coloring of a graph $G$ is unfriendly if every vertex has at least as many neighbors of the color opposite to his own. Such a coloring always exist and can be obtained by partitionin …
- 3,463
3
votes
1
answer
118
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Inertia of the cone graph
Let $\widehat{G}$ be the graph obtained by adding a vertex to a graph $G$ and joining it to all vertices in $V(G)$. Let $\sigma(G)$ be the number of non-positive eigenvalues of the adjacency matrix of …
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5
votes
1
answer
307
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A variant of Ramsey numbers
The well known Ramsey number $R(k)$ is the least integer $n$ so that every 2-edge coloring of $K_n$ contains a monochromatic $K_k.$
Another interpretation of the above definition is that every graph …
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1
vote
0
answers
76
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Proper edge colorings with no bi-colored 5-paths
Consider you want to properly edge color a graph such that it has no bi-colored cycle. Denote by $\alpha'(G)$ the least number of colors required to color the edges of $G$ in such a way.
It is well k …
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8
votes
2
answers
349
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Isomorphism problem on the class of induced subgraphs of a hypercube
A problem that I am currently studying translates to the problem of deciding whether two induced subgraphs of the hypercube $Q_k$ are isomorphic.
Now it feels to me that this class of graphs is "too …
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7
votes
1
answer
403
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Induced subgraphs of small strongly regular graphs
Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed $ …
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4
votes
4
answers
1k
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Determine if a graph has a large clique
This question is quite specific and practical. I hope it is still relevant for MO and will not be removed.
I have a collection $\mathcal{C}$ of graphs having from 5000-6000 vertices and edge density …
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2
votes
3
answers
240
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Regular graphs whose neighbourhoods induce matchings
Studying some problem I've arrived to the following notion.
Let a $2r$-regular graph $G$ be called neighbour-matching if $N(v) = rK_2.$ In other words, the neighbourhood of any vertex induces a matc …
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5
votes
0
answers
319
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Graphs with many positive eigenvalues of their distance matrix
Let $G$ be a simple connected graph $D(G)$ its distance matrix and $n_{+}(G), n_{-}(G)$ the number of positive and negative eigenvalues of $D(G)$ respectively.
We call a graph $G$ optimistic if $n_{+ …
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6
votes
2
answers
854
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Minimal graphs of prescribed girth and chromatic number
The well known result of Erdős, states that
Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$
What I am wondering is
When can we expect …
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6
votes
1
answer
685
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Probability that a random edge coloring of the complete graph is proper
This is a repost of this math.se question that I am posting here since it received no attention there.
What is the probability that a random edge coloring of $K_n$ with
$m \geq n$ colors resul …
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14
votes
1
answer
887
views
Tutte polynomials, graph complements and degree sequences
Harary and Akiyama asked whether there exists a non self-complementary (SC) graph $G$ having the same chromatic polynomial as its complement.
It was later shown that there indeed exist such graphs a …
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33
votes
10
answers
6k
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Is the empty graph a tree?
This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed.
The que …
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4
votes
1
answer
575
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Generating non-isomorphic graphs by adding edges to a given graph
This question is in a way related to the one I posted on math.se. Since the question there did not produce any final answer I am trying my luck here!
I am given a fairly large graph $G$ and subsets $ …
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2
votes
0
answers
275
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Structure of almost all bipartite graphs
I am studying some properties related to bipartite graphs and it would be useful for me to know if there is anything known about the structure of almost all bipartite graphs. For example, is it true t …
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