All Questions
Tagged with cayley-graphs co.combinatorics
25
questions
16
votes
1
answer
507
views
Chromatic numbers of infinite abelian Cayley graphs
The recent striking progress on the chromatic number of the plane by de Grey arises from the interesting fact that certain Cayley graphs have large chromatic number; namely, the graph whose vertices ...
- 19.1k
5
votes
0
answers
189
views
Groups of non-orientable genus 1 and 2
The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar ...
- 150
5
votes
0
answers
155
views
When is a Schreier coset graph vertex transitive
When is a Schreier Coset graph on a group $G$ with subgroup $H$ and symmetric generating set $S$(without identity) vertex transitive?
It is well known that when $H$ is normal, the Schreier coset graph ...
- 1,841
4
votes
2
answers
416
views
Transposition Cayley graphs are planar
Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
- 1,841
4
votes
1
answer
249
views
Diameter of Cayley graphs of finite simple groups
Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article).
THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
- 225
4
votes
1
answer
166
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Cliques in Cayley graph on $n$-cycles
Let $S\subset S_n$ be the set of all $n$-cycles. I want to know if the Cayley graph $(S_n,S)$ has large dense subgraphs. I'm expecting it to not have super-polynomial size and $1-o(1)$ dense subgraphs....
- 173
4
votes
1
answer
226
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Total coloring conjecture for Cayley graphs
The total coloring conjecture (TCC) states that any total coloring of a simple graph $G(V,E)$ has its total chromatic number bounded as $\chi^{T}(G)\le \Delta+2$ where $\Delta $ is the maximal degree ...
- 1,841
3
votes
1
answer
95
views
Edge coloring of a graph on alternating groups
Let $G$ be the Cayley graph on the alternating group $A_n\,n\ge4$ with generating set $$S=\begin{cases}\{(1,2,3),(1,3,2),\\(1,2,\ldots,n),(1,n,n-1,\ldots,2)\}, &n\ \text{odd}\\ \{(1,2,3),(1,3,2),\\...
- 1,841
3
votes
0
answers
63
views
Hamiltonian cycles in Cayley graph on alternating group
Let $G=\operatorname{Cay}(A_n,S)$ be the Cayley graph on the Alternating group $A_n\quad n\ge4$ with generating set $S=\{(1,2,3),(1,2,4),\ldots,(1,4,2),(1,3,2)\}$. One Hamiltonian cycle in $G$ for $n=...
- 1,841
3
votes
0
answers
211
views
Growth functions of finite group - computation, typical behaviour, surveys?
Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour:
Rubik's growth in LOG scale (see MO322877):
S_n n=9 growth and nice fit by normal ...
- 22.9k
2
votes
1
answer
90
views
Imposing reciprocity in the definition of vertex-transitivity
A simple, undirected graph is vertex-transitive if for any pair of vertices $x,y$, there exists an automorphism (adjacency-preserving self-bijection) $\phi$ such that $\phi(x)=y$.
What if, instead of ...
- 170
2
votes
0
answers
59
views
Path-covering for vertex-transitive graphs
I have the following dummy problem:
Claim - There exists $N$ such that for $n > N$, if $G_n$ be a connected directed vertex-transitive graph with $n$ vertices, then there exists a set $S$ of paths ...
- 2,874
1
vote
2
answers
121
views
Difference in chromatic number between Schreier coset graphs and Cayley graphs
Can the Schreier coset graphs can be seen as a subgraph of Cayley graph on the same groups(neglecting the loop edges) and, hence, have their chromatic numbers bounded by the chromatic numbers of the ...
- 1,841
1
vote
1
answer
221
views
Cayley graphs do not have isolated maximal cliques
Let a Cayley graph $G$ of a group $H$ with respect to the generating set $\{s_i\}$ have a clique of order $> 2$. In addition assume the graph $G$ is non-complete. If the clique size is less than ...
- 1,841
1
vote
0
answers
46
views
Circulant graphs chromatically dominated by powers of cycles
Suppose we can color the vertices of powers of cycles $C_n^k$ using $c$ colors such that each of the color classes $c_i$ have $v_i$ number of vertices. Can we always color the circulant of degree $2k$ ...
- 1,841
1
vote
0
answers
110
views
Chromatic number of certain graphs with high maximum degree
Let $G$ be the graph of even order $n$ and size $\ge\frac{n^2}{4}$ which is a Cayley graph on a nilpotent group but not complete. Can the chromatic number of this graph be determined in polynomial ...
- 1,841
1
vote
0
answers
150
views
Are all even regular undirected Cayley graphs of Class 1?
Are even order Cayley graphs of Class 1, that is, can they be edge-colored with exactly $m$ colors, where $m$ is the degree of each vertex?
I think yes, because of the symmetry the Cayley graphs ...
- 1,841
0
votes
2
answers
134
views
coloring infinite vertex transitive graph without large cliques
Let $G$ be an infinite vertex-transitive graph (this means that for every $u,w \in V(G)$ there exists an automorphism $\tau$ of $G$ such that $\tau(u) = v$).
We assume that $G$ is undirected, and does ...
- 11.2k
0
votes
1
answer
110
views
Bound on chromatic number of graphs on any finite $p$-group
Is the chromatic number of a Cayley graph on $p$-groups with any generating set bounded by the chromatic number of the maximal induced circulant subgraph?
I think yes. Because for one, the main ...
- 1,841
0
votes
1
answer
78
views
Extending the vertex coloring of circulant graph to graph on $p$-group
Let $G_1$ be a circulant graph of prime order $p$. This implies that $G_1$ is the Cayley graph on $\mathbb{Z}_p$ with some generating set $S_1$. I am interested in knowing the characterizations of the ...
- 1,841
0
votes
1
answer
332
views
A vertex transitive graph has a near perfect/ matching missing an independent set of vertices
Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
- 1,841
0
votes
0
answers
112
views
Procedure to color the edges of a circulant graph
From the first theorem in this paper, it is clear that a cayley graph on abelian group for all generating sets of even order is class $1$, that is can be edge colored in exactly $\Delta$ colors. But, ...
- 1,841
0
votes
1
answer
111
views
Recognizing perfect Cayley graphs as tensor products
It is known (and can easily be seen) that a unitary Cayley graph on $n=\prod_ip_i$, ($p_i$ distinct primes) vertices with $n$ square-free can be recognized as the tensor product of the graphs $K_{p_i}$...
- 1,841
-1
votes
1
answer
168
views
Which line graphs of Cayley graphs are Cayley
When are the line graphs of Cayley graphs Cayley?
From this link we can know when the line graphs of complete graphs are cayley. But, my question pertains to the larger class of Cayley graphs. Are ...
- 1,841
-1
votes
1
answer
195
views
Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques
Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
- 1,841