Bounding the Number of Graphs Containing Very Long Induced Paths

• Main Author: Butler, Steven Kay
• Format: Others
• Published: BYU ScholarsArchive 2003
• Subjects: mathematics; combinatorics; graph theory; paths; induced paths; asymptotic behavior; stirling numbers; polya counting; Burnsides theorem; Mathematics
• Online Access: https://scholarsarchive.byu.edu/etd/31; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1030&context=etd

Induced graphs are used to describe the structure of a graph, one such type of induced graph that has been studied are long paths. In this thesis we show a way to represent such graphs in terms of an array with two colors and a labeled graph. Using this representation and the techniques of Polya counting we will then be able to get upper and lower bounds for graphs containing a long path as an induced subgraph. In particular, if we let P(n,k) be the number of graphs on n+k vertices which contains P_n, a path on n vertices, as an induced subgraph then using our upper and lower bounds for P(n,k) we will show that for any fixed value of k that P(n,k)~2^(nk+k_C_2)/(2k!).