Research Statement
ILKYOO CHOI ([email protected])
My area of research is discrete mathematics and combinatorics, with a focus on graph coloring.
I am also interested in Ramsey Theory and extremal questions in both graph theory and combinatorics. In particular, I am intrigued by substructures that must be contained in combinatorial
objects satisfying predetermined conditions.
The theory of coloring plays a central role in not only graph theory but also in all of discrete
mathematics, as it deals with the fundamental problem of partitioning a set of objects into classes
that avoid certain conflicts. Arguably the most famous result in graph coloring is the Four Color
Theorem, which states that the chromatic number of a planar graph is at most four. Since planar
graphs can be characterized via forbidden structures, the Four Color Theorem demonstrates how
forbidding certain structures can bound the chromatic number. I am fascinated by this phenomena
of the relation between forbidden structures and graph coloring parameters. I worked on this
phenomena for the chromatic number in [9, 14] as well as for the list chromatic number, which
is a strengthening of the chromatic number, in [8]. In the process, I provided evidence for both
Geelen’s Conjecture and Gyárfás’s Conjecture, and disproved a conjecture of Cai, Wang, and Zhu.
I [7, 10, 15, 16] also worked on this phenomena for a weakening of the chromatic number known as
improper coloring, where I solved questions by Raspaud and Montassier and Ochem.
Ramsey Theory is a field of investigating substructures that a sufficiently large structure
must contain. The cornerstone result in the area is Ramsey’s Theorem, which states that a host
graph with sufficiently many vertices must contain either a large independent set or a large clique.
In online Ramsey Theory, which is a strengthening of Ramsey Theory, we [6] characterized when
a 3-cycle must appear in the host graph without a fixed subgraph, except for one case. Outside of
graph theory, I [11] worked on a Ramsey Theoretic type question regarding tiling the integers with
the same gap sequence, and this is the first evidence towards a conjecture by Nakamigawa.
My other interests involve extremal questions, which studies how global properties influence
local structures of the object under consideration. I [12, 13, 17] have sharp results where the
extremal parameter in interest is the maximum average degree, the edge density, or the length of
a minimum cycle of a graph in order to obtain the structure we seek. In particular, I answered
a question by Raspaud and Wang applied to graphs embeddable on a torus. I also worked on
combinatorics on words, where we [3] completely determined the minimum number of squares an
infinite partial word must contain.
Below is a brief summary of some of my results and future research plans.
Obstacles in Classical Graph Coloring
A simple lower bound on the chromatic number is the maximum number of pairwise adjacent
vertices. A natural question to ask is if the chromatic number can be upper bounded by a function
f of this lower bound. In general, such an f does not exist by a construction of Erdős [20]. However,
Geelen (see [19]) and Gyárfás [25] made two different conjectures stating that different types of
substructures are obstacles to the existence of such a function f . With Kwon and Oum [14], we
proved that an infinite class of graphs satisfies Geelen’s conjecture, and as a corollary we obtained
new evidence to the aforementioned conjecture of Gyárfás.
Ilkyoo Choi, page 1/5
As mentioned before, the Four Color Theorem [1, 2] states that every planar graph is 4-colorable.
There is a vast literature on identifying obstacles for a planar graph to be 3-colorable. A celebrated
theorem by Grötzsch [23] says planar graphs with no 3-cycles are 3-colorable; he actually proved
that any 3-coloring of either a 4-cycle or a 5-cycle extends to the whole graph. This stemmed
the project of finding obstacles of extending a 3-coloring of a k-cycle to a 3-coloring of the whole
graph. With Ekstein, Holub, and Lidický [9], we pushed the project further and characterized
when a precoloring of a 9-cycle extends to the entire graph. The proof involves some computer
programming to reduce massive case analysis.
List coloring, also known as choosability, is a strengthening of proper coloring. Instead of letting
all colors be available at every vertex, list coloring also considers the situation when different vertices
are allowed to have different colors available. As an approach to the Four Color Theorem, in 1890,
Heawood proved an analogue to the Four Color Theorem to graphs that are embeddable on a
surface of any given Euler genus. Heawood’s bound also works for list coloring.
Cai, Wang, and Zhu [5] conjectured that Heawood’s bound can be improved when certain
substructures are forbidden in graphs embeddable on a torus. However, I [8] constructed an infinite
family of counterexamples, where the conjecture is false even in a stronger sense of list coloring.
Moreover, I proved a weakening of the conjecture, which was suggested by one of the originators of
the conjecture. Even though the conjecture is false for graphs that are embeddable on a torus, a
natural direction is to consider graphs that are embeddable on other surfaces. The counterexample
I constructed actually works for all surfaces except the plane and the projective plane. Since the
question is already solved for planar graphs, I plan to investigate the truth of the question for
projective planar graphs.
Improper Graph Coloring
A relaxed version of proper coloring is improper coloring. In this version of coloring, some prescribed
defects are allowed in each color class, where defects are measured in terms of the maximum degree
of the graph induced by the vertices of a color class. As this is a natural question, research on
improper coloring planar graphs started even before the term improper coloring was first defined.
It is known that for any pair (d1 , d2 ), there exists a planar graph with minimum cycle length
4 such that the vertex set cannot be partitioned into two parts where each part induces a graph
with maximum degree at most d1 and d2 . Hence, many researchers tried to determine all pairs
(d1 , d2 ) where the vertex set of every planar graph with minimum cycle length at least 5 can be
partitioned into two parts with maximum degree at most d1 and d2 . My first contribution in the
area is in [16], where Raspaud and I solved one additional pair, which solved a question of Raspaud.
Later, with Choi, Jeong, and Suh [7], we provided infinitely many more pairs, answering a question
of Montassier and Ochem [26]; moreover, only finitely many values remain to be determined.
In this vein, with Esperet [10], we extended many results regarding proper coloring planar
graphs, including the Four Color Theorem and Grötzsch’s Theorem, to results that involve graphs
that are embeddable on a surface with any given Euler genus. In addition, all of our results are
tight in every way possible as we construct explicit examples.
All aforementioned results concern the length of a shortest cycle as a parameter in order to
obtain positive results. However, one may ask if it is necessary to forbid all short cycles. With Liu
and Oum [15], we characterized all cycle obstruction sets for improper coloring planar graphs, for
any given number of parts. This means that we identified all inclusionwise minimal sets S of cycles
Ilkyoo Choi, page 2/5
that guarantee planar graphs without cycles in S can be improperly colored. I plan to investigate
the cycle obstruction sets for graphs on a surface with given Euler genus in the future.
Ramsey Theory
Ramsey Theory is a phenomena that says a nice substructure exists even when there is no immediate
structure given. The foundational result is Ramsey’s Theorem [28], which states that in a host
graph with sufficiently many vertices, there exists a subset of the vertices that are all pairwise
either adjacent or nonadjacent.
Online Ramsey Theory is a strengthening of Ramsey Theory, where instead of revealing the
entire host graph at once in order to find the substructure in question, small parts of the host
graph are revealed gradually. Restricting the host graph to a certain class of graphs was first
done by Grytczuk, Kierstead, and Pralat [24] where they investigated when the target graph is a
3-cycle. With Choi, Jeong, and Oum [6], we initiated the study of online Ramsey Theory where
a small subgraph is forbidden in the host graph. In particular, we extend a result in [24], and
we characterized when a 3-cycle must appear in the host graph that does not contain a particular
subgraph, except for one case.
Ramsey Theory is a phenomena that not only appears in graph theory, but also shows up in
many other areas of discrete mathematics. Recently, with Jung and Kim [11], we worked on a
Ramsey Theoretic type question regarding partitioning the integers into sets with the same gap
sequence. This was a question by Nakamigawa [27], and we are the first ones to give positive evidence. A natural generalization of tiling the integers is tiling the d-dimensional integer lattice. The
analogue of gap sequences to higher dimensions is the Euclidean actions. Tiling the d-dimensional
integer lattice has many open questions, and I would like to start approaching them one at a time.
Extremal Questions
Extremal combinatorics studies extremal objects that satisfy certain properties. Extremality can
be taken with respect to different invariants such as the number of vertices, the number of edges,
and the length of a shortest cycle. More abstractly, it studies how global properties influence local
structures of the object under consideration. For example, with Kim, Tebbe, and West [13], we
proved a sharp threshold on the density of a (hyper)graph to contain several sub(hyper)graphs that
cover the same part of the graph the same number of times.
With Kim, Kostochka, and Raspaud [12], we showed that the edge set of sparse graphs with
large maximum degree can be partitioned so that each part satisfies a distance requirement. In
particular, we show that the number of parts required drastically changes as the maximum average
degree changes. With Zhang [17], we found a sharp threshold on the length of a shortest cycle of
graphs embeddable on a torus that guarantees the vertex set of the graph can be partitioned into
two parts where each part induces a graph with no cycles. This completely answered a question by
Raspaud and Wang [29] applied to graphs embeddable on a torus.
I am also interested in extremal properties of objects not in graph theory. The field of combinatorics on words concerns strings of letters. The cornerstone result in this area is by Thue [31, 32],
who proved that there exists an infinite binary word without cubes and an infinite ternary word
without squares. Fraenkel and Simpson [22] constructed an infinite binary word with only three
squares, and Dekking [18] constructed an infinite binary word without cubes with only ten squares,
Ilkyoo Choi, page 3/5
and both bounds on the number of squares are tight. With Blanchet-Sadri and Mercaş [3], we
extended both aforementioned results to partial words, which are words where some letters are unknown and hence are treated as wild cards. In the future, I would like to push the project further
for words over a larger alphabet.
Future plans
My current interests lie in structures that must be present in combinatorial objects satisfying
a predetermined condition. I will continue investigating this phenomena in the area of graph
coloring, Ramsey Theory, and extremal questions in all of discrete mathematics. I am broadening
my horizon by working on a project involving characterizing minor obstructions for generalizations
of outerplanar graphs. As tools in discrete mathematics can be utilized in other areas, it is also a
great opportunity for interdisciplinary research. For example, I am currently involved in a project
in discrete geometry.
In the area of graph coloring, Brooks’ Theorem [4] is a classical result characterizing when the
chromatic number cannot be improved from the greedy bound. For graphs with large maximum
degree, Reed [30] strengthened this result. With Kierstead, Rabern, and Reed, I am involved in
a project that extends Reed’s result from the chromatic number to the list coloring setting. It
would be interesting to see if we are able to go further and generalize the result to correspondence
coloring, which is a new strengthening of list coloring.
As for improper coloring, I am interested to see what stronger restrictions we can impose in
addition to bounding the maximum degree of each color class. One historical generalization would
be bounding the size of a component in each color class. To obtain such a coloring, Esperet and
Ochem [21] considered minimum cycle length conditions for graphs embeddable on a surface of any
given Euler genus. I plan to investigate the cycle obstruction sets for this generalization for planar
graphs first, and then move on to graphs embeddable on a surface of any given Euler genus.
I am currently working in a new area of research. We are trying to characterize the minor
obstructions for generalizations of outerplanar graphs. One generalization is outer-toroidal graphs,
which are graphs that are embeddable on the torus where all vetices are incident with one face.
Another generalization is k-outerplanar graphs, which are planar graphs where all vertices are
incident with at most k faces. If we succeed, then I believe the result will aid the understanding of
graphs embeddable on the torus.
References
[1] K. Appel and W. Haken. Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429–490,
1977.
[2] K. Appel, W. Haken, and J. Koch. Every planar map is four colorable. II. Reducibility. Illinois J. Math.,
21(3):491–567, 1977.
[3] F. Blanchet-Sadri, I. Choi, and R. Mercaş. Avoiding large squares in partial words. Theoret. Comput. Sci.,
412(29):3752–3758, 2011.
[4] R. L. Brooks. On colouring the nodes of a network. Proc. Cambridge Philos. Soc., 37:194–197, 1941.
[5] L. Cai, W. Wang, and X. Zhu. Choosability of toroidal graphs without short cycles. J. Graph Theory, 65(1):1–15,
2010.
[6] H. Choi, I. Choi, J. Jeong, and S.-i. Oum. Online Ramsey theory for a triangle on F -free graphs. Submitted,
2016.
Ilkyoo Choi, page 4/5
[7] H. Choi, I. Choi, J. Jeong, and G. Suh. (1, k)-Coloring of Graphs with Girth at Least Five on a Surface. Journal
of Graph Theory, to appear, 2016.
[8] I. Choi. Toroidal graphs containing neither K5 nor 6-cycles are 4-choosable. Journal of Graph Theory, to appear,
2016.
[9] I. Choi, J. Ekstein, P. Holub, and B. Lidický. 3-coloring triangle-free planar graphs with a precolored 9-cycle.
Submitted, 2016.
[10] I. Choi and L. Esperet. Improper coloring of graphs on surfaces. Submitted, 2016.
[11] I. Choi, J. Jung, and M. Kim. On tiling the integers with 4-sets of the same gap sequence. Submitted, 2016.
[12] I. Choi, J. Kim, A. V. Kostochka, and A. Raspaud. Strong edge-colorings of sparse graphs with large maximum
degree. Submitted, 2016.
[13] I. Choi, J. Kim, A. Tebbe, and D. B. West. Equicovering subgraphs of graphs and hypergraphs. Electron. J.
Combin., 21(1):Paper 1.62, 17, 2014.
[14] I. Choi, O-j. Kwon, and S.-i. Oum. Coloring graphs without fan vertex-minors and graphs without cycle pivotminors. J. Combinatorial Theory Ser. B, to appear, 2016.
[15] I. Choi, C.-H. Liu, and S.-i. Oum. Characterization of cycle obstruction sets for improper coloring planar graphs.
Submitted, 2016.
[16] I. Choi and A. Raspaud. Planar graphs with girth at least 5 are (3, 5)-colorable. Discrete Math., 338(4):661–667,
2015.
[17] I. Choi and H. Zhang. Vertex arboricity of toroidal graphs with a forbidden cycle. Discrete Math., 333:101–105,
2014.
[18] F. M. Dekking. On repetitions of blocks in binary sequences. J. Combinatorial Theory Ser. A, 20(3):292–299,
1976.
[19] Z. Dvořák and D. Král. Classes of graphs with small rank decompositions are χ-bounded. European J. Combin.,
33(4):679–683, 2012.
[20] P. Erdős. Graph theory and probability. Canad. J. Math., 11:34–38, 1959.
[21] L. Esperet and P. Ochem. Islands in graphs on surfaces. SIAM J. Discrete Math., 30(1):206–219, 2016.
[22] A. S. Fraenkel and R. J. Simpson. How many squares must a binary sequence contain? Electron. J. Combin.,
2:Research Paper 2, approx. 9 pp. (electronic), 1995.
[23] H. Grötzsch. Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel.
Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe, 8:109–120, 1958/1959.
[24] J. A. Grytczuk, H. A. Kierstead, and P. Pralat. On-line Ramsey numbers for paths and stars. Discrete Math.
Theor. Comput. Sci., 10(3):63–74, 2008.
[25] A. Gyárfás. Problems from the world surrounding perfect graphs. In Proceedings of the International Conference
on Combinatorial Analysis and its Applications (Pokrzywna, 1985), volume 19, pages 413–441 (1988), 1987.
[26] M. Montassier and P. Ochem. Near-colorings: non-colorable graphs and NP-completeness. Electron. J. Combin.,
22(1):Paper 1.57, 13, 2015.
[27] T. Nakamigawa. One-dimensional tilings using tiles with two gap lengths. Graphs Combin., 21(1):97–105, 2005.
[28] F. P. Ramsey. On a Problem of Formal Logic. Proc. London Math. Soc., S2-30(1):264.
[29] A. Raspaud and W. Wang. On the vertex-arboricity of planar graphs. European J. Combin., 29(4):1064–1075,
2008.
[30] B. Reed. A strengthening of Brooks’ theorem. J. Combin. Theory Ser. B, 76(2):136–149, 1999.
[31] A. Thue. Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiana, 7:1–22, 1906.
(Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, Norway
(1977), pp. 139–158).
[32] A. Thue. Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat.
Nat. Kl. Christiana, 1:1–67, 1912. (Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor,
Universitetsforlaget, Oslo, Norway (1977), pp. 413–478).
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