Bounds on Maximum Matchings in 1-Planar Graphs

Graph Matchings;1-Planar Graphs

Wittnebel, Johnadvisor:Biedl, Therese ; affiliation1:Faculty of Mathematics ; Biedl, Therese ;

University of Waterloo

关键词: 1-Planar Graphs;    Graph Matchings;    Master Thesis;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/14445/3/Wittnebel_John.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

In this thesis, we study lower bounds on maximum matchings in 1-planar graphs. We expand upon the tools used for proofs of matching bounds in other classes of graphs as well as some original ideas in order to find these bounds.The first novel results we provide are lower bounds on maximum matchings in 1-planargraphs as a function of their minimum degree. We show that for sufficiently large n, 1-planargraphs with minimum degree 3 have a maximum matching of size at least (n+12)/7, 1-planar 7graphs with minimum degree 4 have a maximum matching of size at least (n+4)/3, and 1-planar 3graphs with minimum degree 5 have a maximum matching of size at least (2n+3)/5. All of these 5bounds are tight. We also give examples of 1-planar graphs with small maximum matching and minimum degree 6 and 7. We conjecture that the 1-planar graph of minimum degree 6 presented has the smallest maximum matching over all 1-planar graphs of minimum degree 6, but it is unclear if the method used for the cases of minimum degree 3, 4, and 5 would work for minimum degree 6.We also study lower bounds in the class of maximal 1-plane graphs, and 3-connectedmaximal 1-plane graphs. We find that 3-connected, maximal 1-plane graphs have a maximummatching of size at least (n+4)/3, and that maximal 1-plane graphs have a maximum matching 3of size at least (n+6)/4. Again, we present examples of such a graph to show this bound is tight. 4We also show that every simple 3-connected maximum 1-planar graph has a matching of sizeat least (2n+6)/5, and provide some evidence that this is tight.