Conflict-free coloring of graphs

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page

Complements of nearly perfect graphs

A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A. Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$, we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $\chi$-bounded

Why do colours look the way they do?

A major part of the mind–body problem is to explain why a given set of physical processes should give rise to perceptual qualities of one sort rather than another. Colour hues are the usual example considered here, and there is a lively debate as to whether the results of colour vision science can provide convincing explanations of why colours actually look the way they do. The internal phenomenological structure of colours is considered here in some detail, and a comparison is drawn with sounds and their synthesis. This paper examines the type of explanation that is needed, and it is concluded that it does not have to be reductive to be effective. What needs to be explained more than anything is why inverted hue scenarios are more intuitive than other sensory inversions: and the issue of physicalism versus dualism is argued to be of only marginal relevance