χ-bounded

In graph theory, a ${\displaystyle \chi }$-bounded family ${\displaystyle {\mathcal {F}}}$ of graphs is one for which there is some function ${\displaystyle c}$ such that, for every integer ${\displaystyle t}$ the graphs in ${\displaystyle {\mathcal {F}}}$ with no ${\displaystyle t}$-vertex clique can be colored with at most ${\displaystyle c(t)}$ colors. This concept and its notation were formulated by András Gyárfás.[1] The use of the Greek letter chi in the term ${\displaystyle \chi }$-bounded is based on the fact that the chromatic number of a graph ${\displaystyle G}$ is commonly denoted ${\displaystyle \chi (G)}$.

Nontriviality

It is not true that the family of all graphs is ${\displaystyle \chi }$-bounded. As Zykov (1949) and Mycielski (1955) showed, there exist triangle-free graphs of arbitrarily large chromatic number,[2][3] so for these graphs it is not possible to define a finite value of ${\displaystyle t(3)}$. Thus, ${\displaystyle \chi }$-boundedness is a nontrivial concept, true for some graph families and false for others.[4]

Specific classes

Every class of graphs of bounded chromatic number is (trivially) ${\displaystyle \chi }$-bounded, with ${\displaystyle c(t)}$ equal to the bound on the chromatic number. This includes, for instance, the planar graphs, the bipartite graphs, and the graphs of bounded degeneracy. Complementarily, the graphs in which the independence number is bounded are also ${\displaystyle \chi }$-bounded, as Ramsey's theorem implies that they have large cliques.

Vizing's theorem can be interpreted as stating that the line graphs are ${\displaystyle \chi }$-bounded, with ${\displaystyle c(t)=t}$.[5][6] The claw-free graphs more generally are also ${\displaystyle \chi }$-bounded with ${\displaystyle c(t)\leq (t-1)^{2}}$. This can be seen by using Ramsey's theorem to show that, in these graphs, a vertex with many neighbors must be part of a large clique. This bound is nearly tight in the worst case, but connected claw-free graphs that include three mutually-nonadjacent vertices have even smaller chromatic number, ${\displaystyle c(t)=2(t-1)}$.[5]

Other ${\displaystyle \chi }$-bounded graph families include:

• The perfect graphs, with ${\displaystyle c(t)=t-1}$
• The graphs of boxicity two[7]
• The graphs of bounded clique-width[8]
• The intersection graphs of scaled and translated copies of any compact convex shape in the plane[9]
• The circle graphs, and (generalizing circle graphs) the "outerstring graphs", intersection graphs of bounded curves in the plane that all touch the unbounded face of the arrangement of the curves[10]

However, although intersection graphs of convex shapes, circle graphs, and outerstring graphs are all special cases of string graphs, the string graphs themselves are not ${\displaystyle \chi }$-bounded. They include as a special case the intersection graphs of line segments, which are also not ${\displaystyle \chi }$-bounded.[4]

Unsolved problems

Unsolved problem in mathematics: Are all tree-free graph classes ${\displaystyle \chi }$-bounded? (more unsolved problems in mathematics)

According to the Gyárfás–Sumner conjecture, for every tree ${\displaystyle T}$, the graphs that do not contain ${\displaystyle T}$ as an induced subgraph are ${\displaystyle \chi }$-bounded. For instance, this would include the case of claw-free graphs, as a claw is a special kind of tree. However, the conjecture is known to be true only for certain special trees, including paths[1] and radius-two trees.[11]

Another unsolved problem on ${\displaystyle \chi }$-bounded was posed by Louis Esperet, who asked whether every hereditary class of graphs that is ${\displaystyle \chi }$-bounded has a function ${\displaystyle c(t)}$ that grows at most polynomially as a function of ${\displaystyle t}$.[6]

Unsolved problem in mathematics: In a hereditary ${\displaystyle \chi }$-bounded graph class, is the chromatic number at most polynomial in the clique size? (more unsolved problems in mathematics)

References

1. Gyárfás, A. (1987), "Problems from the world surrounding perfect graphs", Proceedings of the International Conference on Combinatorial Analysis and its Applications (Pokrzywna, 1985), Zastosowania Matematyki, 19 (3–4): 413–441 (1988), MR 0951359
2. Zykov, A. A. (1949), "О некоторых свойствах линейных комплексов" [On some properties of linear complexes], Mat. Sbornik N.S. (in Russian), 24 (66): 163–188, MR 0035428. Translated into English in Amer. Math. Soc. Translation, 1952, MR0051516. As cited by Pawlik et al. (2014)
3. Mycielski, Jan (1955), "Sur le coloriage des graphs", Colloq. Math. (in French), 3: 161–162, MR 0069494
4. Pawlik, Arkadiusz; Kozik, Jakub; Krawczyk, Tomasz; Lasoń, Michał; Micek, Piotr; Trotter, William T.; Walczak, Bartosz (2014), "Triangle-free intersection graphs of line segments with large chromatic number", Journal of Combinatorial Theory, Series B, 105: 6–10, arXiv:1209.1595, doi:10.1016/j.jctb.2013.11.001, MR 3171778
5. Chudnovsky, Maria; Seymour, Paul (2010), "Claw-free graphs VI. Colouring", Journal of Combinatorial Theory, Series B, 100 (6): 560–572, doi:10.1016/j.jctb.2010.04.005, MR 2718677
6. Karthick, T.; Maffray, Frédéric (2016), "Vizing bound for the chromatic number on some graph classes", Graphs and Combinatorics, 32 (4): 1447–1460, doi:10.1007/s00373-015-1651-1, MR 3514976
7. Asplund, E.; Grünbaum, B. (1960), "On a coloring problem", Mathematica Scandinavica, 8: 181–188, doi:10.7146/math.scand.a-10607, MR 0144334
8. Dvořák, Zdeněk; Král', Daniel (2012), "Classes of graphs with small rank decompositions are ${\displaystyle \chi }$-bounded", Electronic Journal of Combinatorics, 33 (4): 679–683, arXiv:1107.2161, doi:10.1016/j.ejc.2011.12.005, MR 3350076
9. Kim, Seog-Jin; Kostochka, Alexandr; Nakprasit, Kittikorn (2004), "On the chromatic number of intersection graphs of convex sets in the plane", Electronic Journal of Combinatorics, 11 (1), R52, MR 2097318
10. Rok, Alexandre; Walczak, Bartosz (2014), "Outerstring graphs are ${\displaystyle \chi }$-bounded", Proceedings of the Thirtieth Annual Symposium on Computational Geometry (SoCG'14), New York: ACM, pp. 136–143, doi:10.1145/2582112.2582115, MR 3382292
11. Kierstead, H. A.; Penrice, S. G. (1994), "Radius two trees specify ${\displaystyle \chi }$-bounded classes", Journal of Graph Theory, 18 (2): 119–129, doi:10.1002/jgt.3190180203, MR 1258244

External links

• Chi-bounded, Open Problem Garden