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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 …

Is it possible to $3$-color the elements of the symmetric group $S_n\ n\ge3$ such that all color classes have the same number of elements ($\frac{n!}{3}$); and, when elements in any color class are a …

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 …

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 …

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 …

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 …

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, …

The Alon-Tarsi number is the least number $k$ such that the coefficient with degree $d$ of the graph polynomial $P(G)=\prod\limits_{i<j}(x_i-x_j)$,( where $x_i$ corresponds to a vertex and a term $x_i …

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{ …

Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A_n$ using the generating set $\{(1,2,3),(1,2,4),\dotsc,(1,2,n),(1,n,2),\dotsc,(1,4,2),(1,3,2)\}$. Note t …

A proper coloring of the vertices of a graph $G$ is seen as a homomorphism from the graph vertices to the complete graph on the number of vertices equal to the chromatic number of the graph. Similarl …

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 C …

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 obstru …

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 …

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 …