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

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...

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...

The Maximum Number of Edges in a Strongly Multiplicative Graph

k-forested choosability of graphs with bounded maximum average degree

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prov...