The Kneser graph \( K(m, n) \) (when \( m > 2n \)) has the \( n \)-subsets of an \( m \)-set as its vertices, two vertices being adjacent in \( K(m, n) \) whenever they are disjoint sets. The \( k \)th-chromatic number of any graph \( G \) (denoted by \( \chi_k(G) \)) is the least integer \( t \) such that the vertices can be assigned \( k \)-subsets of \( \{1, 2, \dots, t\} \) with adjacent vertices receiving disjoint \( k \)-sets. S. Stahl has conjectured that, if \( k = qn – r \) where \( q \geq 1 \) and \( 0 \leq r < n \), then \( \chi_k(K(m, n)) = qm – 2r \). This expression is easily verified when \( r = 0 \); Stahl has also established its validity for \( q = 1 \), for \( m = 2n + 1 \) and for \( k = 2, 3 \). We show here that the expression is also valid for all \( q \geq 2 \) in the following further classes of cases:

1. \( 2n + 1 < m \leq n(2 + r^{-1}) \) (\( 0 < r 1 \));
2. \( 4 \leq n \leq 6 \) and \( 1 \leq r \leq 2 \) (all \( m \));
3. \( 7 \leq n \leq 11 \) and \( r = 1 \) (all \( m \));
4. \( (n, r, m) = (7, 2, 18), (12, 1, 37), (12, 1, 38) \) or \( (13, 1, 40) \).

We prove that for a finite planar space \( S = (\mathcal{P}, \mathcal{L}, \mathcal{H}) \) with no disjoint planes and with a constant number of planes on a line, the number \( \ell \) of lines is greater than or equal to the number \( c \) of planes, and the equality holds true if and only if \( S \) is either the finite Desarguesian 4-dimensional projective space \( PG(4,q) \), or the complete graph \( K_5 \).

A \((p, q)\)-graph \( G \) is said to be \(\textbf{edge graceful}\) if the edges can be labeled by \( 1, 2, \ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. J. Mitchem and A. Simoson introduced the concept of super edge-graceful graphs, which is a stronger concept than edge-graceful for some classes of graphs.

A graph \( G = (V, E) \) of order \( p \) and size \( q \) is said to be \(\textbf{super edge-graceful}\) (SEG) if there exists a bijection
\[
\text{f: E} \to
\begin{cases}
\{0, +1, -1, +2, -2, \ldots, \frac{q-1}{2}, -\frac{q-1}{2}\} & \text{if } q \text{ is odd} \\
\{+1, -1, +2, -2, \ldots, \frac{q}{2}, -\frac{q}{2}\} & \text{if } q \text{ is even}
\end{cases}
\]
such that the induced vertex labeling \( f^* \) defined by \( f^*(u) = \sum \{ f(u, v) : (u, v) \in E \} \) has the property:
\[
f^*: V \to
\begin{cases}
\{0, +1, -1, \ldots, +\frac{p-1}{2}, -\frac{p-1}{2}\} & \text{if } p \text{ is odd} \\
\{+1, -1, \ldots, +\frac{p}{2}, -\frac{p}{2}\} & \text{if } p \text{ is even}
\end{cases}
\]
is a bijection.

The conjecture is still unsettled. In this paper, we first characterize spiders of even orders which are not SEG. We then exhibit some spiders of even orders which are SEG of diameter at most four. By the concepts of the irreducible part of an even tree \( T \), we show that an infinite number of spiders of even orders are SEG. Finally, we provide some conjectures for further research.