Finished justification of linear combination of G_1 patterns contributing to G_3 three-matchings.

poly-form.tex

@@ -269,9 +269,10 @@ For Tri, note that it is the case that the graph $G_2$ is a 'triangle of two pat
 
 \qed
 
-In a similar way we can count the number of 3-matchings in graph $G_3$, where each edge in a given $G_1$ gets replaced with a disjoint 3-path, disjoint meaining that no other 3-path intersects another 3-path, except at its endpoints as in the original graph.  The linear combination for 3-matchings in $G_3$ follows.
+In a similar way we can count the number of 3-matchings in graph $G_3$, where each edge in a given $G_1$ gets replaced with a disjoint 3-path, disjoint meaining that no other 3-path intersects another 3-path, except at its endpoints as in the original graph.  Because of $G_3$ construction, we now need to also account for two paths in $G_1$.  
+
+The linear combination for 3-matchings in $G_3$ follows.
 
-Because of $G_3$ construction, we now need to also account for two paths in $G_1$.
 
 
 \begin{align*}
@@ -279,4 +280,17 @@ Because of $G_3$ construction, we now need to also account for two paths in $G_1
 &+ 40\pbrace{\numocc{G_1}{\twopathdis}} + 32\pbrace{\numocc{G_1}{\oneint}} + 45\pbrace{\numocc{G_1}{\threedis}}
 \end{align*}
 
-\AH{Justification next.}
+\AH{Justification next.}
+Enumerate through the RHS in a similar fashion.  Beginning with a two path $\twopath$ in $G_1$, it is the case that in $G_3$ this becomes a six-path.  As discussed previously, this yields four three matching subgraphs.  For subgragh of two disjoint edges, $\twodis$, this becomes two disjoint 3-paths.  It is the case in one 3-path, that we have one subgraph of two disjoint edges, where a third disjoint edge can be picked from any of the three edges in the remaining disjoint 3-path.  The process can be repeated starting with the alternative 3-path, giving $2 * 3$ unique 3-matchings.
+
+Now for the 3-edge subgraphs, starting with a triangle.  When a triangle in $G_1$ is transformed into $G_3$, it becomes a 'triangle' where each leg is a three-path.  This is very similar to a 9-path, with the caveat that the first and last edge cannot be in the same 3-matching set together.  Iterating through all possible combinations producing 3-matchings, i.e. $(e_1, e_3, e_5),\ldots, (e_1, e_3, e_8), (e_1, e_4, e_6),\ldots, (e_1, e_4, e_8),$\newline$\ldots, (e_1, e_6, e_8),\ldots, (e_5, e_7, e_9)$ gives a total of 
+
+\[\sum_{i = 1}^4i+ \sum_{i = 1}^4i + \sum_{i = 1}^3i + \sum_{i = 1}^2i + 1 = 30\]
+
+matchings.  Consider next a 3-path in $G_1$, where the resulting subgraph in $G_3$ is a 9-path.  In this case, because the endpoints are disconnected, we have five other 3-matchings that couldn't be counted in the case of the Tri subgraph, namely $(e_1, e_3, e_9),\ldots, (e_1, e_7, e_9)$, thus $30$ (from the Tri analysis)$ + 5 = 35$ three-matchings.
+
+ For the $\twopathdis$ subgraph, it is the case that this graph becomes a 6-path with a disjoint 3-path in $G_3$.  We have to consider 3 possibilities of 3-matchings.  First, the 6-path produces 4.  Second, it is the case that the 6-path produces 10 two-matchings, which can be paired with any one of the three edges in the disjoint 3-path, producing $10 * 3$ 3-matchings.  Third, the disjoint 3-path can produce one 3-matching which can be paired with a third edge from any one of the edges in the 6-path, giving $1 * 6 = 6$ three-matchings, for a total of $4 + 30 + 6 = 40$ three-matchings.
+
+Given the $Fan$ subgraph, where 3 distinct edges are connected at one common endpoint, occurring in $G_1$.  In $G_3$, this becomes 3 distinct 9-paths, with each 9-path intersecting the others at one common and shared endpoint.  If we consider the outermost non-intersecting edges along with the middle non-intersecting edges, we have $2 * 2 * 2 = 8$ possible 3-matchings.  Considering the inner, intersecting edges, we have the condition that only one can appear at a time in a 3-matching set.  When we pick an arbitrary inner edge, we have one of two possibilities, we can pick the outer edge of the same 3-path the inner edge is located on, while picking any of the other 4 remaining edges in the middle and outer edges of the other two 3-paths.  This gives $4 * 3$ more unique 3-matchings.  The remaining possibility exists in combining the arbitrary inner edge with any of the 4 combinations of the middle and outer edges of the other 3-paths.  This yields again $3 * 4 = 12$ unique three-matchings, together which make $8 + 12 + 12 = 32$ three-matchings.
+
+Given the $\threedis$ subgraph occurring in $G_1$, the resulting graph consists of three disjoint 3-paths in $G_3$.  There are two considerations.  First, if we pull one edge from each disjoint 3-path, we have three choices from each path, which is $3^3 = 27$ three-matchings.  The second consideration is that we can pull a two matching from any of the given disjoint 3-paths, matching it with a third disjoint edge from any of the other edges in the other 2 three-paths, giving $3 * 6 = 18$ more unique 3-matchings for a total of $27 + 18 = 45$ 3-matchings.