A triple of natural numbers (a,b,c) is an S-set if a+b=c. I. Schur used the S-sets to show that for n __>__3, there exists s(n) such that for prime p > s(n), x^p + y^p = z^p (mod p) has a nontrivial solution. A (p,q)-graph G is said to be vertex Ho-Lee-Schur graph if there exists a bijection f: V(G) –> {1,2,…,p} such that for each C3 subgraph of G with vertices {x,y,z} the triple (f(x),f(y),f(z)) is an S-set. The VHLS deficiency of G is the smallest k such that GU Nk, where Nk is null graph, is a vertex Ho-Lee-Schur graph. We determine VHLS deficiency of some graphs and show that no Kuratowski type characterization of non-vertex Ho-Lee-Schur graphs. Some relation of integer partitions and this theory is explored. We will also introduce some unsolved problems and invite the audience to solve them.

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