Ilkyoo Choi

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A k,d-list assignment L of a graph G is a function that assigns to each vertex v a list Lv of at least k colors and for any adjacent pair xy, the lists Lx and Ly share at most d colors. A graph G is k,d-choosable if there exists an L-coloring of G for every k,d-list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are 3, 1-choosable and that planar graphs without 5- and 6-cycles are 3, 1-choosable. In addition, we give an alternative and slightly stronger proof that triangle-free planar graphs are 3, 1-choosable.

Planar graphs with girth at least 5 are ( 3 , 5 ) -colorable

A graph is ( d 1 , ? , d r ) -colorable if its vertex set can be partitioned into r sets V 1 , ? , V r where the maximum degree of the graph induced by V i is at most d i for each i ? { 1 , ? , r } . Let G g denote the class of planar graphs with minimum cycle length at least g . We focus on graphs in G 5 since for any d 1 and d 2 , Montassier and Ochem constructed graphs in G 4 that are not ( d 1 , d 2 ) -colorable. It is known that graphs in G 5 are ( 2 , 6 ) -colorable and ( 4 , 4 ) -colorable, but not all of them are ( 3 , 1 ) -colorable. We prove that graphs in G 5 are ( 3 , 5 ) -colorable, leaving two interesting questions open: (1) are graphs in G 5 also ( 3 , d 2 ) -colorable for some d 2 ? { 2 , 3 , 4 } ? (2) are graphs in G 5 indeed ( d 1 , d 2 ) -colorable for all d 1 + d 2 ? 8 where d 2 ? d 1 ? 1 ?

(1, k)-Coloring of Graphs with Girth at Least Five on a Surface

A graph is (d1,...,dr)-colorable if its vertex set can be partitioned into r sets V1,...,Vr so that the maximum degree of the graph induced by Vi is at most di for each i∈{1,...,r}. For a given pair (g,d1), the question of determining the minimum d2=d2(g,d1) such that planar graphs with girth at least g are (d1,d2)-colorable has attracted much interest. The finiteness of d2(g,d1) was known for all cases except when (g,d1)=(5,1). Montassier and Ochem explicitly asked if d2(5, 1) is finite. We answer this question in the affirmative with d2(5,1)≤10; namely, we prove that all planar graphs with girth at least five are (1, 10)-colorable. Moreover, our proof extends to the statement that for any surface S of Euler genus γ, there exists a K=K(γ) where graphs with girth at least five that are embeddable on S are (1, K)-colorable. On the other hand, there is no finite k where planar graphs (and thus embeddable on any surface) with girth at least five are (0, k)-colorable.

Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

A fan

Strong edge-colorings of sparse graphs with large maximum degree

F_k

3-Coloring Triangle-Free Planar Graphs with a Precolored 9-Cycle

is a graph that consists of an induced path on

Avoiding large squares in partial words

k

Characterization of Cycle Obstruction Sets for Improper Coloring Planar Graphs

vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers

On tiling the integers with 4-sets of the same gap sequence

q

Toroidal graphs containing neither K5− nor 6-cycles are 4-choosable

and