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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

2 votes
0 answers
119 views

Alon Tarsi reaches its lower bound for complete multipartite graphs

Consider a complete multipartite graph $G$ with $k$ $(k\ge2)$ parts with equal number of vertices $n$, where $n$ is even. If $k=2$, it is a well known result that the Alon-Tarsi number (ATN)(The maxim …
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1 vote
Accepted

Three coloring the elements of symmetric group

Yes, it is possible to $3$-color the elements of the Symmetric group in the way stated. The proof can be found in Theorem 1 of MDPI Symmetry Paper. The proof uses a similar idea given in the comments …
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0 votes
0 answers
54 views

Clique sizes of generalized Kneser graphs

Are there known bounds for clique size in generalized Kneser graphs $KG(n,k,t)=K(n,k,t-1)$, the graph formed by distinct $k$ subsets of $n$ set so that two subsets with at most $t$ elements in common …
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0 votes
0 answers
56 views

Cycles in Kneser graphs with three vertices forming triangles

Consider the Kneser graphs $G=K(n,k)$. Is it possible to list how many even cycles, or, at the least, existence of an even cycle of a given order in $G$, such that any three consecutive vertices form …
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0 votes
0 answers
50 views

Determining homomorphism using automorphism group of two graphs

I wish to know the connection between the automorphism group of two graphs and homomorphism between them, if any. Like all Kneser graphs $K(n,k)$ have the same automorphism group $S_n$. But, given ano …
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1 vote
1 answer
240 views

Choosing sets with a few properties from a given set of elements

Fix $n$ and $k$ with $n \geq 2k+1$. Let $X$ be an $n$ element set. Let $\binom{X}{k}$ denote the collection of $k$-element subsets of $X$. Suppose that $\mathcal{Y} \subseteq \binom{X}{k}$ is a family …
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0 votes
0 answers
51 views

Kernel perfection in some powers of cycles

Suppose I orient the edges of the power of cycle graph $G=C_n^k$ where $n=16$ and $k=4$ in such a way that all the generated cycles by the elements $1,2,3,4$ are given the standard lexical orientation …
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1 vote
0 answers
134 views

Misunderstanding the definition of kernel in digraphs

By Borodin–Kostochka–Woodall '97 paper, the first paragraph says that directed odd cycles do not have kernels. But, I don't get this. Like, consider any $\lfloor\frac{n}{2}\rfloor$ set of independent …
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2 votes
2 answers
140 views

Regarding a specific Turán number of graphs

I wish to know the latest bound on the number of edges a graph of girth greater than or equal to $t$ can have. Specifically, I heard somewhere that a graph of girth greater than or equal to $t$ can ha …
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3 votes
1 answer
100 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),\\(2, …
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0 votes
0 answers
107 views

Bound on the number of maximum matchings in a graph

It is known that the number of perfect matchings in a graph is bounded above by the integer part of the square root of the permanent of its adjacency matrix. But, suppose I take the square root of the …
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0 votes
1 answer
77 views

Line graphs of complete hypergraphs as complement of Kneser graphs

Since the Johnson graph/triangular graph $J(n,2)$ is the complement of the Kneser graph $K(n,2)$, which is also incidentally the line graph of the complete graph $K_n$, I thought whether the same can …
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2 votes

Difference between Alon-Tarsi number and the list chromatic number

Let us consider the graph $K_{n,n}$. Since the graph polynomial of $K_{n,n}$ is homogenous with degree $n^2$, we must have the Alon-Tarsi number of the graph $K_{n,n}$ to be $\ge \frac{n^2}{2n}=\frac{ …
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3 votes
0 answers
72 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= …
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1 vote
1 answer
202 views

Words representations of elements of a symmetric group

Let $S=\{(1,2),(1,2,\ldots,n),(1,n,,n-1,\ldots,2)\}$ be a subset of the symmetric group $S_n$. Let $a=(1,2),b=(1,2,\ldots,n),c=(1,n,n-1\ldots,2)$ be the elements of $S$. My question is, since $S$ is a …
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