5 research outputs found
Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
Extremal and Ramsey Type Questions for Graphs and Ordered Graphs
In this thesis we study graphs and ordered graphs from an extremal point of view. In the first part of the thesis we prove that there are ordered forests H and ordered graphs of arbitrarily large chromatic number not containing such H as an ordered subgraph. In the second part we study pairs of graphs that have the same set of Ramsey graphs. We support a negative answer to the question whether there are pairs of non-isomorphic connected graphs that have this property. Finally we initiate the study of minimal ordered Ramsey graphs. For large families of ordered graphs we determine whether their members have finitely or infinitely many minimal ordered Ramsey graphs