Let *p* be a positive real number and S(*p*) the circle in the
plane with circumference *p*. A subset U of S(*p*) is said to be *assignable*
if it is the union of finitely many open arcs on S(*p*). Let length(U)
denote the length of U. If G is a graph, a function L that assigns to each
vertex *v* of G an assignable subset L(*v*) of S(*p*) is called a
list assignment. An L-coloring of G is a function *c* from V(G) to S(*p*)
such that *c*(*v*) is an element of L(*v*) for each vertex *v*
of G and such that for every pair *u*, *v* of adjacent vertices of G,
the distance between *c*(*u*) and *c*(*v*) on S(*p*) is
at least 1.

The graph G is *circularly* *t*-*choosable* if for every *p*
and every list assignment L in S(*p*) with length(L(*v*)) greater or
equal to *t* for every vertex *v*, there exists an L-coloring of G.

There are many questions that one can ask about circular choosability. Let us mention some of them:

**Problem 1:**
Is it true that a graph with list chromatic number *k*
is circularly *k*-choosable?

**Problem 2:**
If G is a graph with maximum vertex degree D, is the line
graph of G circularly (D+1)-choosable?

**Problem 3:**
Which cubic graphs are circularly *t*-choosable for
some *t* that is smaller than 4?

**Problem 4:**
What is the smallest real number *t* such that every
planar graph is circularly *t*-choosable?

Most likely, the answer to Problem 4 is between 4 and 5.

Send comments to Bojan.Mohar@uni-lj.si