Rainbow numbers for small stars with one edge added

• Autorzy: Izolda Gorgol , Ewa Łazuka
• Języki publikacji: EN

Abstrakty

EN

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f(n,H) is the maximum number of colors in an edge-coloring of Kₙ with no rainbow copy of H. The rainbow number rb(n,H) is the minimum number of colors such that any edge-coloring of Kₙ with rb(n,H) number of colors contains a rainbow copy of H. Certainly rb(n,H) = f(n,H) + 1. Anti-Ramsey numbers were introduced by Erdös et al. [5] and studied in numerous papers.
We show that $rb(n,K_{1,4} + e) = n + 2$ in all nontrivial cases.

Słowa kluczowe

EN

rainbow number; anti-Ramsey number

Kategorie tematyczne

• 05C35: Extremal problems
• 05C55: Generalized Ramsey theory

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2010
• Tom: 30
• Numer: 4
• Strony: 555-562

Daty

wydano

2010

otrzymano

2008-12-29

poprawiono

2009-10-30

zaakceptowano

2009-10-30

Twórcy

autor

Izolda Gorgol

• Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38D, 20-618 Lublin, Poland

autor

Ewa Łazuka

• Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38D, 20-618 Lublin, Poland

Bibliografia

• [1] N. Alon, On the conjecture of Erdös, Simonovits and Sós concerning anti-Ramsey theorems, J. Graph Theory 7 (1983) 91-94, doi: 10.1002/jgt.3190070112.
• [2] M. Axenovich and T. Jiang, Anti-Ramsey numbers for small complete bipartite graphs, Ars Combinatoria 73 (2004) 311-318.
• [3] R. Diestel, Graph theory (Springer-Verlag, New York, 1997).
• [4] P. Erdös and M. Simonovits, A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966) 51-57.
• [5] P. Erdös, A. Simonovits and V. Sós, Anti-Ramsey theorems, in: Infinite and Finite Sets (A. Hajnal, R. Rado, and V. Sós, eds.), Colloq. Math. Soc. J. Bolyai (North-Holland, 1975) 633-643.
• [6] I. Gorgol, On rainbow numbers for cycles with pendant edges, Graphs and Combinatorics 24 (2008) 327-331, doi: 10.1007/s00373-008-0786-8.
• [7] T. Jiang, Anti-Ramsey numbers for subdivided graphs, J. Combin. Theory (B) 85 (2002) 361-366, doi: 10.1006/jctb.2001.2105.
• [8] T. Jiang, Edge-colorings with no large polychromatic stars, Graphs and Combinatorics 18 (2002) 303-308, doi: 10.1007/s003730200022.
• [9] T. Jiang and D.B. West, On the Erdös-Simonovits-Sós conjecture about the anti-Ramsey number of a cycle, Combin. Probab. Comput. 12 (2003) 585-598, doi: 10.1017/S096354830300590X.
• [10] T. Jiang and D.B. West, Edge-colorings of complete graphs that avoid polychromatic trees, Discrete Math. 274 (2004) 137-145, doi: 10.1016/j.disc.2003.09.002.
• [11] J.J. Montellano-Ballesteros, Totally multicolored diamonds, Electronic Notes in Discrete Math. 30 (2008) 231-236, doi: 10.1016/j.endm.2008.01.040.
• [12] J.J. Montellano-Ballesteros and V. Neuman-Lara, An anti-Ramsey theorem on cycles, Graphs and Combinatorics 21 (2005) 343-354, doi: 10.1007/s00373-005-0619-y.
• [13] I. Schiermeyer, Rainbow 5- and 6-cycles: a proof of the conjecture of Erdös, Simonovits and Sós, preprint (TU Bergakademie Freiberg, 2001).
• [14] I. Schiermeyer, Rainbow numbers for matchings and complete graphs, Discrete Math. 286 (2004) 157-162, doi: 10.1016/j.disc.2003.11.057.
• [15] M. Simonovits and V. Sós, On restricted colorings of Kₙ, Combinatorica 4 (1984) 101-110, doi: 10.1007/BF02579162.

Identyfikatory

DOI

10.7151/dmgt.1513