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Results tagged with graph-theory
Search options questions only
not deleted
user 6094
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.
6
votes
2
answers
653
views
Cut locus in a graph
I am wondering if the concept of a cut locus has been defined and explored in discrete graphs, rather than their usual home on manifolds?
The Wikipedia definition (which I believe I (co-?)authored) is …
- 63.9k
22
votes
2
answers
895
views
Is every 1-million-connected graph rigid in 3D?
It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete Methods 3, no …
- 32.1k
6
votes
5
answers
1k
views
Generate random graphs that satisfy the triangle inequality
I would like to generate random graphs that might be geometric graphs in some
(unknown) dimension. So I would like every triangle in the graph to satisfy the
triangle inequality on its (random) edge l …
- 37.7k
24
votes
6
answers
3k
views
Shortest grid-graph paths with random diagonal shortcuts
Suppose you have a network of edges connecting
each integer lattice point
in the 2D square grid $[0,n]^2$
to each of its (at most) four neighbors, {N,S,E,W}.
Within each of the $n^2$ unit cells of thi …
- 35.5k
5
votes
2
answers
715
views
Bound on graph domination number when min degree is 7
I have a graph $G$ whose minimum vertex degree is $\delta=7$.
I am seeking an upper bound on the domination number $\gamma(G)$
in terms of the number of vertices $n$ of $G$.
I found a paper by
Edwin C …
- 1,446
14
votes
2
answers
730
views
Blinking graphs
For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a bit assignment for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the …
- 32.1k
6
votes
4
answers
546
views
(Non)uniqueness of the common-factor graph
Let $S=\{x_1,\ldots,x_k\}$ be a set of $k$ distinct natural numbers,
a subset of $\{1,\ldots,n\} = \mathbb{N}_{\le n}$.
Define the common-factor graph $G(S)$ as the (undirected) graph with
a node for …
- 32.1k
24
votes
3
answers
2k
views
Gauss-Bonnet Theorem for Graphs?
One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an $n$-cycle has $\chi = 0$ and $K_4$ has $\chi=-2$.
Is there an analog for the Gaus …
- 32.1k
10
votes
2
answers
714
views
Bounds on chromatic number of $k$-planar graphs
A $1$-planar graph can be drawn in the
plane so that each arc is crossed at most once by another arc.
A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times.
Planar graphs are …
- 32.1k
8
votes
2
answers
591
views
Orthogonal Hamiltonian cycles in (n x n x n) grids
Let $C_n$ be a cubical $n \times n \times n$ subset of the integer lattice,
so consisting of $n^3$ vertices.
I am interested in special Hamiltonian cycles in $C_n$, special in the
sense that (a) each …
- 1,444
25
votes
2
answers
2k
views
Who first dubbed them "expander graphs"?
Expander graphs
("sparse graphs that have strong connectivity properties")
burst onto the mathematical scene around the millennium, but I have not
been successful in tracing the origin of
(a) the conc …
- 91
5
votes
0
answers
130
views
Equitable 4-colorings of planar triangulations
In an
equitable coloring
of a graph $G$, the number of vertices in each color class differ
by at most $1$.
For example, left below is not an equitable coloring, while the
right graph is equitably colo …
- 151k
16
votes
4
answers
1k
views
Squaring a square and discrete Ricci flow
Is this a theorem?
Every $3$-connected planar graph $G$ may be represented as
a tiling of a square by squares,
one square per node of $G$, with nodes connected in $G$
corresponding to tangent squares …
- 151k
3
votes
0
answers
159
views
Rubber-band graph embedding under gravity
Let $G$ be a simple $3$-connected plane graph with the vertices of
its outer face pinned to the $xy$-plane.
View each edge as an ideal rubber band, and each internal
node given a weight proportional t …
- 151k
16
votes
3
answers
2k
views
Are infinite planar graphs still 4-colorable?
Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has a …
- 32.5k