Search Results

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 …

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 …

Consider the graph $G$ of order $n$ consisting of two disjoint cliques of even order $\frac{n}{2}=p+1$ (where $p$ is odd prime) joined by a bipartite graph (that is, deleting the edges of the two di …

The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $n\times n$ array, its elements could be filled exactly like a latin square, where the elements in …

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 …

Consider a regular graph of order $n$ and degree $\Delta$. Now, by Brooks' theorem, we can partition the vertices into $\Delta+1$ independent sets. The extreme case of $n$ independent sets is only for …

The Evans conjecture ( which was proved later by Smetaniuk) states that for any $n$, if at most $n-1$ entries of a partial $n\times n$ latin square are filled, it can be completed to the full latin s …

What is an upper bound on the chromatic number of the square of a tree on $n$ vertices? Note that the power of the graph is considered in this sense. If the tree were a path, then it is easy to see th …

There exist inequalities relating the maximum and minimum eigenvalues of the adjacency matrix of a graph with its chromatic numbers, i.e. the Wilf's and Hoffmann's inequalities, which put together sta …

It is known that there exist polynomial time algorithm to approximate the Lovasz number or the supremum of Shannon capacity of graphs. By Vizing's theorem, the graph $G$ has only two chromatic indic …

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, …

Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ co …

Consider any graph with $n$ vertices and maximum degree $\Delta$. By Vizing's theorem, the graph could be edge colored(properly) with at most $\Delta+1$ colors. My question pertains as to what the ma …

Consider the graph formed by $k$ cliques of order $k$, any two cliques sharing at most one point in common. Now, by Szekeres-Wilf theorem, I think the graph should be $k$ colorable, as any connected i …

Consider the Total Graph ($T(G)$) of a graph $G$ with vertex set $V$ edge set $E=\{(u,v)\}$ with Line graph $L(G)$ and subdivision graph $S(G)$ (formed by putting a vertex in each edge of the original …