Landon Rabern

A Note On Reed's Conjecture

In [J. Graph Theory, 27 (1998), pp. 177-212], Reed conjectures that every graph satisfies

On hitting all maximum cliques with an independent set

\chi \leq \lceil \frac{\omega + \Delta + 1}{2} \rceil

Dangerous Reference Graphs and Semantic Paradoxes

. We prove that this holds for graphs with disconnected complement. Combining this fact with a result of Molloy proves the conjecture for graphs satisfying

List-Coloring Claw-Free Graphs with

\chi > \lceil\frac{n}{2}\rceil

\Delta-1

. Generalizing this we prove that the conjecture holds for graphs satisfying

Colors

\chi > \frac{n + 3 - \alpha}{2}

Destroying Noncomplete Regular Components in Graph Partitions

. It follows that the conjecture holds for graphs satisfying

Coloring Graphs with Dense Neighborhoods

\Delta \geq n + 2 - (\alpha + \sqrt{n + 5 - \alpha})

Coloring a graph with Δ - 1 colors

. In the final section, we show that if

A game generalizing Hall's Theorem

G