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Holbrook, Hobert J.

Skaggs, R. Duane

English

Morehead State University

Forcing number, maximum nullity, and minimum rank theorems concerning the generalized Petersen graph and related observations*Hobert J. Holbrook* Dr. R. Duane Skaggs Mathematics DepartmentIntroductionWhat is Z(P(n,k))?We know that Z(P(n,k))≤n. Below is the proof: Pf: Given a Petersen graph with the interior subgraph which is a cycle graph, color that cycle graph’s vertices. Then, each vertex will force one vertex, so n vertices are forced since the cycle subgraph has n vertices. So, since this coloring is sufficient but not necessarily necessary, we have an upperbound on the forcing number. It helps to have a picture to really understand this proof. Let’s look at the standard Petersen graph:As the arrows indicate, the red vertices force the remaining blue vertices. The star subgraph is a cycle graph. So, in the general case, we color the cycle subgraph (whatever it is) and it forces the “perimeter vertices”. An alternative approach to establishing this is coloring the perimeter, which will force the interior. Either way works. All in all, though, this begs the question: Does Z(P(n,k))=n? Here are some examples that would make one think this is the case:Through brute force, or some slightly technical calculations of related parameters, we can establish the fact that all three of these graphs having a forcing number of n for each case of n. The first has a forcing number of 11, the second has a forcing number of 7, and the third has a forcing number of 9. In all these cases, Z(P(n,k))=n. However, other generalized Petersen graphs exhibit different patterns. For example, look at these generalized Petersen graphs:So, is the forcing number for the generalized Petersen graphs always n? Don’t these graphs which exhibit different patterns suggest there is more to the story?Let v and u be vertices of a graph G. We define the following coloring rule: If a vertex v and all its neighbors except u are colored, then v forces u to be colored. The forcing number Z(G), or simply Z, of a graph is the minimum number of colored vertices which forces the entire graph to be colored. We can think of the generalized Petersen graph, denoted by P(n,k), as two graphs connected together. First, the graph in the interior is the circulant graph Ci_n(k), which is, in most cases, a cycle on n vertices. (A cycle is a graph that always looks like a polygon on n vertices.) The graph on the perimeter is always a cycle, namely a cycle on n vertices. Each vertex in the perimeter is adjacent to one and only one vertex in the interior, and no vertex in the perimeter is adjacent to the same vertex in the interior as another vertex in the perimeter. This is best explained pictorially:Here the interior graph (the square) is a cycle on 4 vertices and the exterior graph (also a square) is the same cycle on 4 vertices. We also see that the vertices are connected on the exterior to the interior are connected in pairs (the top left corner on the outside is connected to the top left corner on the inside, for example). Now that we understand what a generalized Petersen graph is, here is an example of a forcing set and an example of a nonforcing set (the graph in this example is P(4,1)):The colored vertices on the left graph force the entire graph since each vertex only has 1 uncolored neighbor and there are 4 colored vertices and 4 uncolored vertices. On the other hand, the right graph’s colored vertices do not force anything since they all have more than 1 uncolored neighbor.Is Z(P(n,k))=n?Many examples would lead us to conclude that Z(P(n,k))=n. In fact, it seems for any graph where n/k is not a positive integer, we have that Z(P(n,k))=n. In these cases, there is a single cycle in the interior of the generalized Petersen graph. However, counterexamples exist. In fact, we’ve already seen one: P(6,3). Here is a coloring of 5 vertices, not 6, which forces P(6,3):Here is how this set of red vertices forces the entire graph: 1. The red vertices force the golden vertices. 2. The red vertices and the golden vertices force the purple vertex. 3. The red vertices, the golden vertices, and the purple vertex force the green vertices. The union of all these colored vertices is the entire vertex set of the graph, so the entire graph is colored. So, this begs the question: What is Z(P(n,k))? Why is it n sometimes and not n other times?ConjectureWe conjecture that C_(n/k) is a subgraph of P(n,k), and that there are k copies of this subgraph when n/k is a positive integer. We conjecture that, once this can be proven, we can get an idea of what Z(P(n,k)) is based on these cyclic subgraphs. This bizarre behavior of Z being not equal to n seems to only occur when the interior graph has more than one cyclic graph composing it. Hence, if we could prove this conjecture, we could then pose a theorem for Z(P(n,k)) in general based on this structure theorem. We call this conjecture the “Star Polygon Conjecture”, since the interior graph is in its entirety known as a star polygon.AcknowledgementsI dedicate this work to: * M. Faye Holbrook and Mary Lou Holbrook, * Bradis Lafferty, * Chris Leadford, * and Isaac Vinson. Without their love and support, I would be much less of what I am today.                                                                 Signed,                                                                  H. John Holbrook.

Forcing number, maximum nullity, and minimum rank theorems concerning the generalized Petersen graph and related observations

https://scholarworks.moreheadstate.edu/cgi/viewcontent.cgi?article=1081&amp;context=student_scholarship_posters

Forcing number, maximum nullity, and minimum rank theorems concerning the generalized Petersen graph and related observations

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