FAW 2019: Frontiers in Algorithmics pp 111-120

# Vertex-Critical ($$P_5$$, banner)-Free Graphs

• Qingqiong Cai
• Shenwei Huang
• Tao Li
• Yongtang Shi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11458)

## Abstract

Given two graphs $$H_1$$ and $$H_2$$, a graph is $$(H_1,H_2)$$-free if it contains no induced subgraph isomorphic to $$H_1$$ or $$H_2$$. Let $$P_t$$ and $$C_t$$ be the path and the cycle on t vertices, respectively. A banner is the graph obtained from a $$C_4$$ by adding a new vertex and making it adjacent to exactly one vertex of the $$C_4$$. For a fixed integer $$k\ge 1$$, a graph G is said to be k-vertex-critical if the chromatic number of G is k and the removal of any vertex results in a graph with chromatic number less than k. The study of k-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $$(k-1)$$-colorable. In this paper, we show that there are finitely many 6-vertex-critical ($$P_5$$, banner)-free graphs. This is one of the few results on the finiteness of k-vertex-critical graphs when $$k>4$$. To prove our result, we use the celebrated Strong Perfect Graph Theorem and well-known properties on k-vertex-critical graphs in a creative way.

## Notes

### Acknowledgments

Qingqiong Cai was partially supported by National Natural Science Foundation of China (No. 11701297) and Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (No. CICIP2018005). Shenwei Huang is partially supported by the National Natural Science Foundation of China (11801284). Tao Li is partially supported by the National Natural Science Foundation (61872200) and the National Key Research and Development Program of China (2018YFB1003405, 2016YFC0400709). Yongtang Shi was partially supported by National Natural Science Foundation of China (Nos. 11771221, 11811540390), Natural Science Foundation of Tianjin (No. 17JCQNJC00300), and the China-Slovenia bilateral project “Some topics in modern graph theory” (No. 12-6).

## References

1. 1.
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, London (2008)
2. 2.
Bruce, D., Hoàng, C.T., Sawada, J.: A certifying algorithm for 3-colorability of P$$_5$$-free graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 594–604. Springer, Heidelberg (2009).
3. 3.
Chudnovsky, M., Goedgebeur, J., Schaudt, O., Zhong, M.: Obstructions for three-coloring and list three-coloring H-free graphs. arXiv:1703.05684 [math.CO]
4. 4.
Chudnovsky, M., Goedgebeur, J., Schaudt, O., Zhong, M.: Obstructions for three-coloring graphs with one forbidden induced subgraph. In: Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1774–1783 (2016)Google Scholar
5. 5.
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)
6. 6.
Dhaliwal, H.S., Hamel, A.M., Hoàng, C.T., Maffray, F., McConnell, T.J.D., Panait, S.A.: On color-critical ($${P}_5$$, co-$${P_5}$$)-free graphs. Discrete Appl. Math. 216, 142–148 (2017)
7. 7.
Dirac, G.A.: Note on the colouring of graphs. Mathematische Zeitschrift 54, 347–353 (1951)
8. 8.
Dirac, G.A.: A property of 4-chromatic graphs and some remarks on critical graphs. J. Lond. Math. Soc. 27, 85–92 (1952)
9. 9.
Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)
10. 10.
Fouquet, J.L.: A decomposition for a class of $$({P}_5,\overline{P_5})$$-free graphs. Discrete Math. 121, 75–83 (1993)
11. 11.
Gallai, T.: Kritische Graphen I. Publ. Math. Inst. Hungar. Acad. Sci. 8, 165–92 (1963)
12. 12.
Gallai, T.: Kritische Graphen II. Publ. Math. Inst. Hungar. Acad. Sci. 8, 373–395 (1963)
13. 13.
Goedgebeur, J., Schaudt, O.: Exhaustive generation of $$k$$-critical $$\cal{H}$$-free graphs. J. Graph Theory 87, 188–207 (2018)
14. 14.
Hell, P., Huang, S.: Complexity of coloring graphs without paths and cycles. Discrete Appl. Math. 216, 211–232 (2017)
15. 15.
Hoàng, C.T., Kamiński, M., Lozin, V.V., Sawada, J., Shu, X.: Deciding $$k$$-colorability of $$P_5$$-free graphs in polynomial time. Algorithmica 57, 74–81 (2010)
16. 16.
Hoàng, C.T., Moore, B., Recoskiez, D., Sawada, J., Vatshelle, M.: Constructions of $$k$$-critical $${P_5}$$-free graphs. Discrete Appl. Math. 182, 91–98 (2015)
17. 17.
Huang, S., Li, T., Shi, Y.: Critical (P6, banner)-free graphs. Discrete Appl. Math. (2018).
18. 18.
Kaminski, M., Pstrucha, A.: Certifying coloring algorithms for graphs without long induced paths. arXiv:1703.02485 [math.CO] (2017)
19. 19.
Kostochka, A.V., Yancey, M.: Ore’s conjecture on color-critical graphs is almost true. J. Combin. Theory Ser. B 109, 73–101 (2014)
20. 20.
Ore, O.: The Four Color Problem. Academic Press, Cambridge (1967)
21. 21.
Randerath, B., Schiermeyer, I.: $$3$$-colorability $$\in \cal{P}$$ for $$P_6$$-free graphs. Discrete Appl. Math. 136, 299–313 (2004)
22. 22.
West, D.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (2001)Google Scholar

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Qingqiong Cai
• 1
• Shenwei Huang
• 1
• Tao Li
• 1
• Yongtang Shi
• 2
Email author
1. 1.College of Computer ScienceNankai UniversityTianjinChina
2. 2.Center for Combinatorics and LPMCNankai UniversityTianjinChina