On the maximum number of edges in planar graphs of bounded degree and matching number

Maximum number of edges in claw-free graphs whose maximum degree and matching number are bounded

We determine the maximum number of edges that a claw-free graph can have, when its maximum degree and matching number are bounded. This is a famous problem that has been studied on general graphs, and for which there is a tight bound. The graphs achieving this bound contain in most cases an induced copy of K1,3, the claw, which motivates studying the question on claw-free graphs. Note that on g...

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [2] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for seve...

The Maximum Number of Edges in a Strongly Multiplicative Graph

The maximum number of edges in 2K2-free graphs of bounded degree

We call a graph 2K2-free if it is connected and does not contain two independent edges as an induced subgraph. The assumption of connectedness in this definition only serves to eliminate isolated vertices. Wagon [6] proved that x(G) ~ w(G)[w(G) + 1]/2 if G is 2Krfree where x(G) and w(G) denote respectively the chromatic number and maximum clique size of G. Further properties of 2K2-free graphs ...

THE MAXIMUM NUMBER OF EDGES IN x*-FREE GRAPHS OF BOUNDED DEGREE

We call a graph 2K,-free if it is connected and does not contain two independent edges as an induced subgraph. The assumption of connectedness in this definition only serves to eliminate isolated vertices. Wagon [6] proved that x(G)co(G)[o(G)+ 1]/2 if G is 2K,-free where x(G) and w(G) denote respectively the chromatic number and maximum clique size of G. Further properties of 2Kz-free graphs ha...