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Results tagged with graph-theory
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user 9025
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
a necessary condition for a nonempty graph being a line graph
No. The graphs you define are called "quasi-line graphs" and are a larger class than line graphs. If you search for "quasi-line graph" you will find a lot of literature on them.
The simplest counte …
- 37k
4
votes
How to solve the opposite of max flow problem?
There is a technique for allowing both minimum and maximum capacities for each edge, where both the bounds can be positive or negative. It works by converting the network to a normal one where the edg …
- 37k
2
votes
co spanning tree
The paper here uses this name for the complement of a spanning tree, i.e. the set of edges which do not lie in some given spanning tree.
- 37k
1
vote
The terminology for a node's number of in-links in weighted directed graph
I've seen papers where $m$ is called the in-degree and $k_{in}$ is called something else (such as weighted in-degree). Using in-degree and fan-in as Vel Nias suggests would be fine too. But you need …
- 37k
7
votes
Accepted
Reference Request: Graph Edge Density
For cycles of odd length, the only extremal graphs for large $n$ are complete bipartite graphs with the sides as equal as possible. For smaller $n$ there can be other extremal graphs. The complete sto …
- 37k
2
votes
4-regular graph with every edge lying in a unique 4-cycle
The number of vertices $n$ must be even or the number of 4-cycles is not an integer. The number of simple connected quartic graphs with the first condition is 0 for $n<12$ and $2,4,25,459$ for $n=12, …
- 37k
4
votes
Diameter of undirected, connected, vertex-transitive graph on $n$ vertices
The cycle has the greatest diameter. Apart from the cycle, the greatest diameter is close to $n/4$. For example consider a circular ladder consisting of two cycles of length $n/2$ connected by $n/2$ r …
- 37k
4
votes
Bounds on spanning tree for sparse graphs
Two bounds I found in my old files:
Grimmett, Disc Math 16 (1976) 323-324. Two bounds:
$$ \frac{1}{n}\left( \frac{2m}{n-1} \right)^{n-1}.$$
$$ \left( \frac{2m-d}{n-1} \right)^{n-1},$$
where $d$ is t …
- 37k
11
votes
Accepted
Bound on the number of minimal vertex covers of a graph
The union of $k$ triangles has $3^k$ minimum vertex covers. You can easily find connected examples.
- 37k
1
vote
Find all edges not covered by a shortest path in an all-pairs shortest path over a subset of...
Note that you can run the Floyd-Warshall algorithm in $O(|V|^2\,|V'|)$ time to obtain shortest paths from $V'$ to everything. You need a $|V|\times |V'|$ matrix instead of a $|V|\times |V|$ matrix. I …
- 37k
14
votes
Accepted
Genus of a graph
No. The two subgraphs can share the surface more efficiently than that. Take a graph $G$ with genus $g\ge 1$ and duplicate each edge. If you don't like double edges, subdivide them with new vertices …
- 37k
0
votes
Spanning trees of $H \cup e$ in terms of $H$
Here is a simple bound. If $H$ is connected and $e=(v,w)\notin E(H)$, let $d$ be the degree of $v$ in the graph $H+e$. Then $$\kappa(H+e)\le d\kappa(H).$$ Proof: $\kappa(H)/\kappa(H+e)$ is the probabi …
- 37k
2
votes
What graph invariants are fast to compute?
Maybe there is a problem with processing of digraphs in sage. nauty takes 0.07 seconds to canonically label each of the graphs $(17;1,3)$ and $(17;3,1)$. They are indeed isomorphic. This translates t …
- 37k
0
votes
Question about the balance of a signed graph construction
The only connected graphs whose balance is independent of the edge ordering are trees and polygons.
Obviously trees have this property.
A connected graph which is neither a tree nor a polygon contai …
- 37k
4
votes
Accepted
Are there 2-connected regular graphs whose maximum matching leaves 3 vertices uncovered?
Take an even number $r\ge 4$. Take $r$ copies of a 3-connected graph $G$ which has odd order and is regular of degree $r$. Add two new vertices $x$ and $y$. For each copy of $G$, remove one edge an …
- 37k