Title: A generalization of Schönemann's theorem via a graph theoretic method,
Abstract: Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013),
359--398], using tools from additive combinatorics and group theory, proved
necessary and sufficient conditions under which the linear congruence
$a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, where $a_1,\ldots,a_k,b,n$ ($n\geq
1$) are arbitrary integers, has a solution $\langle x_1,\ldots,x_k \rangle \in
\Z_{n}^k$ with all $x_i$ distinct modulo $n$. So, it would be an interesting
problem to give an explicit formula for the number of such solutions. Quite
surprisingly, this problem was first considered, in a special case, by
Schönemann almost two centuries ago(!) but his result seems to have been
forgotten. Schönemann [{\it J. Reine Angew. Math.} {\bf 1839} (1839),
231--243] proved an explicit formula for the number of such solutions when
$b=0$, $n=p$ a prime, and $\sum_{i=1}^k a_i \equiv 0 \pmod{p}$ but $\sum_{i \in
I} a_i \not\equiv 0 \pmod{p}$ for all $I\varsubsetneq \lbrace 1, \ldots,
k\rbrace$. In this paper, we generalize Schönemann's theorem using a result
on the number of solutions of linear congruences due to D. N. Lehmer and also a
result on graph enumeration recently obtained by Ardila et al. [{\it Int. Math.
Res. Not.} {\bf 2015} (2015), 3830--3877]. This seems to be a rather uncommon
method in the area; besides, our proof technique or its modifications may be
useful for dealing with other cases of this problem (or even the general case)
or other relevant problems.