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Results tagged with graph-theory
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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.
9
votes
Girth 5 graphs with diameter 2
No infinite family exists. In fact all graphs with diameter $d$ and girth $2d+1$ have to be regular, and thus are Moore graphs. This was proved in
R. Singleton, "There is no irregular Moore graph", A …
- 85.6k
3
votes
Hoffman singleton conjecture
The suggestion from the other answer is great. I wanted to add that it is not even known whether the Moore graph of degree 57 must contain a single copy of a Petersen graph, let alone a decomposition …
- 85.6k
8
votes
The chromatic number of the union of two graphs
In a similar vein to $\chi^{\ast}(G_n)$, we can define a quantity $\chi^{\ast \ast}(G_n)$ as follows: Suppose you have an abelian group $M$ and a set $S=\{x_1, x_2, \dots, x_n\}\subset M$, such that f …
- 85.6k
19
votes
Accepted
Reference request: Moore graphs
Moore posed this problem to Hoffman at a conference, so it is not in print. Hoffman makes the following remark (from "Selected Papers of Alan Hoffman with Commentary", pp. 367):
After I discussed the …
- 85.6k
2
votes
Anchor sets for lattice polygons: Part I
Consider the set $S_0$ consisting of all points $(x,y)$ such that $(x,y)\in \mathcal D$ but $(x-1,y)$ and $(x,y-1)$ are not in $\mathcal D$. It is clear that $S_0$ satisfies the conditions in the prob …
- 85.6k
6
votes
Accepted
$1$-factorizability for "complete" finite hypergraphs
This is Baranyai's theorem. Other than in Baranyai's original paper you can also find a cool proof in the article "Uniform hypergraphs" by Brouwer and Schrijver which uses max-flow min-cut.
- 85.6k
8
votes
Accepted
Embedding any graph in a regular graph with the same chromatic number
Yes, you can find such a $G_R$ of any degree greater than or equal to the maximum degree of $G$. This is the main theorem of the paper "On regular bipartite-preserving supergraphs" by G. Chartrand and …
- 85.6k
5
votes
Accepted
Bound on the chromatic number of square of bipartite graphs
The maximum degree of $G^2$ for general $G$ is at most $\Delta^2$, so we immetiately get an upper bound $\chi(G^2)\le \Delta^2+1$.
An example that is close to optimal is the incidence graph of the poi …
- 85.6k
7
votes
Accepted
Chromatic number of square of a tree
The particular case of the square of a tree is easy to handle by producing a greedy $(\Delta+1)$-coloring starting from a root vertex and extending. However, much stronger results are known:
The $k$-t …
- 85.6k
13
votes
Accepted
Does every $C_4$-free bipartite graph lies in some finite projective plane?
This is an open problem posed by Erdos in "Some old and new problems in various branches of combinatorics" (see section 6). There hasn't been any substantial progress since then. After posing the ques …
- 85.6k
5
votes
Accepted
Minimal degree in a critical graph
The only $2$-critical graph is $K_2$ and the only $3$-critical graphs are odd cycles, therefore the answer is yes for $k\in \{2,3\}$. For $k\geq 4$ the answer is no.
Let's start with the easy case, $k …
- 85.6k
6
votes
Finding an element of the homology group of a graph which is everywhere nonzero
The terminology you are looking for is Nowhere-Zero Flow. It is actually known that these exist on any bridgeless graph for any $\mathbb Z/n\mathbb Z$ with $n\geq 6$. One of the biggest problems in th …
- 85.6k
11
votes
Accepted
What is known about graphs that permit only one colouring?
It's a bit hard to give a comprehensive answer without knowing exactly what sort of properties you are after, but here is a start. Such graphs are called uniquely colorable graphs (see here and here). …
- 85.6k
2
votes
Bijective operations on finite simple graphs
One can also cook up an involution based on unique factorization of connected graphs under various products. A connected graph has a unique representation as a cartesian product of irreductible graphs …
- 85.6k
7
votes
Accepted
Version of Hall's marriage theorem in arbitrary finite graphs
This is actually a special case of Hall's theorem itself, rather than an extension of it. To each such graph $G$ you can associate a bipartite graph $G'$ with vertex set two copies of $V$, which we ca …
- 85.6k