## Abstract

A graph *G* is *k-critical* if it has chromatic number *k*, but every proper subgraph of *G* is (*k*−1)-colorable. Let *f*
_{
k
}(*n*) denote the minimum number of edges in an *n*-vertex *k*-critical graph. In a very recent paper, we gave a lower bound, *f*
_{
k
}(*n*)≥(*k, n*), that is sharp for every *n*≡1 (mod *k*−1). It is also sharp for *k*=4 and every *n*≥6. In this note, we present a simple proof of the bound for *k*=4. It implies the case *k*=4 of two conjectures: Gallai in 1963 conjectured that if *n*≡1 (mod *k*−1) then \(f_k (n)\tfrac{{(k + 1)(k - 2)n - k(k - 3)}} {{2(k - 1)}}\), and Ore in 1967 conjectured that for every *k*≥4 and \(n \geqslant k + 2,f_k (n + k - 1) = f(n) + \tfrac{{k - 1}} {2}(k - \tfrac{2} {{k - 1}})\). We also show that our result implies a simple short proof of Grötzsch’s Theorem that every triangle-free planar graph is 3-colorable.

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## References

- [1]
O. V. Borodin, A. V. Kostochka, B. Lidicky and M. Yancey: Short proofs of coloring theorems on planar graphs,

*European J. of Combinatorics***36**(2014), 314–321. - [2]
G. A. Dirac: A theorem of R. L. Brooks and a conjecture of H. Hadwiger,

*Proc. London Math. Soc.***7**(1957), 161–195. - [3]
B. Farzad, M. Molloy: On the edge-density of 4-critical graphs,

*Combinatorica***29**(2009), 665–689. - [4]
T. Gallai: Kritische Graphen I,

*Publ. Math. Inst. Hungar. Acad. Sci.***8**(1963), 165–192. - [5]
T. Gallai: Kritische Graphen II,

*Publ. Math. Inst. Hungar. Acad. Sci.***8**(1963), 373–395. - [6]
H. Götzsch: 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**(1958/1959), 109–120 (in German). - [7]
T. R. Jensen and B. Toft:

*Graph Coloring Problems*,*Wiley-Interscience*Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1995. - [8]
T. R. Jensen and B. Toft: 25 pretty graph colouring problems,

*Discrete Math.***229**(2001), 167–169. - [9]
A. V. Kostochka and M. Yancey: Ore’s Conjecture is almost true, submitted.

- [10]
O. Ore:

*The Four Color Problem*, Academic Press, New York, 1967. - [11]
C. Thomassen: A short list color proof of Göotzsch’s theorem,

*J. Combin. Theory Ser. B***88**(2003), 189–192.

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Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research.

Research of this author is partially supported by the Arnold O. Beckman Research Award of the University of Illinois at Urbana-Champaign and from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.”

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Kostochka, A., Yancey, M. Ore’s conjecture for *k*=4 and Grötzsch’s Theorem.
*Combinatorica* **34, **323–329 (2014). https://doi.org/10.1007/s00493-014-3020-x

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### Keywords

- Planar Graph
- Simple Proof
- Chromatic Number
- Proper Coloring
- Graph Coloring Problem