# Arrangeability and Clique Subdivisions

Chapter

## Summary

Let k be an integer. A graph G is k-arrangeable (concept introduced by Chen and Schelp) if the vertices of G can be numbered v 1, v 2, , v n in such a way that for every integer i with 1 ≤ in, at most k vertices among {v 1, v 2, , v i } have a neighbor $$v \in \{ v_{i+1},v_{i+2},\ldots,v_{n}\}$$ that is adjacent to v i . We prove that for every integer p ≥ 1, if a graph G is not 2500(p + 1)8-arrangeable, then it contains a K p -subdivision. By a result of Chen and Schelp this implies that graphs with no K p -subdivision have “linearly bounded Ramsey numbers,” and by a result of Kierstead and Trotter it implies that such graphs have bounded “game chromatic number.”

## References

1. 1.
K. Appel and W. Haken, Every planar map is four colorable, Contemporary Mathematics 98, Providence, RI, 1989.
2. 2.
H. L. Bodlaender, On the complexity of some coloring games, in: Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science Volume 484, Springer, 1991, 30–40.Google Scholar
3. 3.
B. Bollobás and A. Thomason, Highly Linked Graphs, Combinatorica 16 (1996), 313–320.Google Scholar
4. 4.
S. A. Burr and P. Erdős, On the magnitude of generalized Ramsey numbers, Infinite and Finite Sets, Vol. 1, A. Hajnal, R. Rado and V. T. Sós, eds., Colloq. Math. Soc. Janos Bolyai, North Holland, Amsterdam/London, 1975.Google Scholar
5. 5.
G. Chen and R. H. Schelp, Graphs with linearly bounded Ramsey numbers, J. Comb. Theory Ser B, 57 (1993), 138–149.
6. 6.
V. Chvatál, V. Rödl, E. Szemerédi and W. T. Trotter, The Ramsey number of a graph of bounded degree, J. Comb. Theory Ser B. 34 (1983), 239–243.
7. 7.
R. Diestel, Graph Theory, 4th edition, Springer, 2010.Google Scholar
8. 8.
H. A. Kierstead and W. T. Trotter, Planar graph coloring with an uncooperative partner, J. Graph Theory 18 (1994), 569–584.Google Scholar
9. 9.
J. Komlós and E. Szemerédi, Topological cliques in graphs II, Combinatorics, Probability and Computing 5 (1996), 79–90.Google Scholar
10. 10.
E. Szemerédi, Colloquium at Emory University, Atlanta, GA, April 22, 1994.Google Scholar
11. 11.
R. Thomas and P. Wollan, An improved linear edge bound for graph linkages, Europ. J. Combin. 26 (2005), 309–324.Google Scholar