Ohio University Southern

By means of a qualitative research method known as folknography, a concerted effort was made to discern perceptions of math and math education in the rural Midwest. A community that will be referred to as Midville, located in the state of Illinois, was chosen as the target population for this study. The community and surrounding region stands over one hundred miles from the nearest metro complex. The study was conducted in May of 2006. After completion of the initial field work, data were collected, analyzed, and summarized, producing this document.

A graph G is vertex pancyclic if for each vertex $${v \in V(G)}$$, and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle C k such that $${v \in V(C_k)}$$. Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K 1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K 3}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vp s (G), is the least nonnegative integer m such that L m (G) is s-vertex pancyclic. We show that for a given integer s ≥ 0, $$vp_s(G)\le \left\{\begin{array}{[email protected]{\quad}l} \qquad\quad\quad\,\,\,\,\,\,\, l(G)+s+1: \quad {\rm if} \,\, 0 \le s \le 4 \\ l(G)+\lceil {\rm log}_2(s-2) \rceil+4: \quad {\rm if} \,\, s \ge 5 \end{array}\right.$$ And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.