Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 4 (2015), 541-549.

The $\Delta^2$ conjecture holds for graphs of small order

Cole Franks

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

An L(2,1)-labeling of a simple graph G is a function f : V (G) such that if xy E(G), then |f(x) f(y)| 2, and if the distance between x and y is two, then |f(x) f(y)| 1. L(2,1)-labelings are motivated by radio channel assignment problems. Denote by λ2,1(G) the smallest integer such that there exists an L(2,1)-labeling of G using the integers {0,,λ2,1(G)}. We prove that λ2,1(G) Δ2, where Δ = Δ(G), if the order of G is no greater than (Δ2 + 1)(Δ2 Δ + 1) 1. This shows that for graphs no larger than the given order, the 1992 “Δ2 conjecture” of Griggs and Yeh holds. In fact, we prove more generally that if L Δ2 + 1, Δ 1, and

|V (G)| (L Δ)(L 1 2Δ + 1) 1,

then λ2,1(G) L 1. In addition, we exhibit an infinite family of graphs with λ2,1(G) = Δ2 Δ + 1.

Article information

Source
Involve, Volume 8, Number 4 (2015), 541-549.

Dates
Received: 15 February 2013
Revised: 15 April 2013
Accepted: 2 October 2013
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511370909

Digital Object Identifier
doi:10.2140/involve.2015.8.541

Mathematical Reviews number (MathSciNet)
MR3366009

Zentralblatt MATH identifier
1316.05106

Subjects
Primary: 97K30: Graph theory

Keywords
L(2,1)-labeling graph labeling channel assignment

Citation

Franks, Cole. The $\Delta^2$ conjecture holds for graphs of small order. Involve 8 (2015), no. 4, 541--549. doi:10.2140/involve.2015.8.541. https://projecteuclid.org/euclid.involve/1511370909


Export citation

References

  • J. A. Bondy and V. Chvátal, “A method in graph theory”, Discrete Math. 15:2 (1976), 111–135.
  • G. J. Chang and D. Kuo, “The $L(2,1)$-labeling problem on graphs”, SIAM J. Discrete Math. 9:2 (1996), 309–316.
  • P. Erdős, S. Fajtlowicz, and A. J. Hoffman, “Maximum degree in graphs of diameter $2$”, Networks 10:1 (1980), 87–90.
  • J. Fiala, T. Kloks, and J. Kratochvíl, “Fixed-parameter complexity of $\lambda$-labelings”, Discrete Appl. Math. 113:1 (2001), 59–72.
  • D. Gonçalves, “On the ${L}(p,1)$-labelling of graphs”, pp. 81–86 in EUROCOMB '05: combinatorics, graph theory and applications (Berlin, 2005), vol. 5, Elsevier, Amsterdam, 2007.
  • J. R. Griggs and R. K. Yeh, “Labelling graphs with a condition at distance $2$”, SIAM J. Discrete Math. 5:4 (1992), 586–595.
  • A. Hajnal and E. Szemerédi, “Proof of a conjecture of P. Erdős”, pp. 601–623 in Combinatorial theory and its applications, II (Balatonfüred, 1969), North-Holland, Amsterdam, 1970.
  • W. Hale, “Frequency assignment: Theory and applications”, pp. 1497–1514 in Proceedings of the IEEE, vol. 68, IEEE, 1980.
  • F. Havet, B. Reed, and J.-S. Sereni, “Griggs and Yeh's conjecture and $L(p,1)$-labelings”, SIAM J. Discrete Math. 26:1 (2012), 145–168.
  • F. Kárteszi, Introduction to finite geometries, Texts in Advanced Mathematics 2, North-Holland, Amsterdam, 1976.
  • H. A. Kierstead and A. V. Kostochka, “A short proof of the Hajnal–Szemerédi theorem on equitable colouring”, Combin. Probab. Comput. 17:2 (2008), 265–270.
  • H. A. Kierstead, A. V. Kostochka, M. Mydlarz, and E. Szemerédi, “A fast algorithm for equitable coloring”, Combinatorica 30:2 (2010), 217–224.
  • H. V. Kronk, “Variations on a theorem of Pósa”, pp. 193–197 in The many facets of graph theory, edited by G. Chartrand and S. F. Kapoor, Lecture Notes in Math. 110, Springer, Berlin, 1969.
  • L. Lovász, “Three short proofs in graph theory”, J. Combinatorial Theory Ser. B 19:3 (1975), 269–271.
  • F. S. Roberts, 1988. private communication to J. R. Griggs.
  • D. Sakai, 1991. private communication to J. R. Griggs.