DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 26(1) (2006) 19-21
DOI: 10.7151/dmgt.1297

AN ANTI-RAMSEY THEOREM ON EDGE-CUTS

Juan José Montellano-Ballesteros

Instituto de Matemáticas, U.N.A.M.
Ciudad Universitaria, Coyoacán 04510
México, D.F. México
e-mail: juancho@math.unam.mx

Abstract

Let G = (V(G), E(G)) be a connected multigraph and let h(G) be the minimum integer k such that for every edge-colouring of G, using exactly k colours, there is at least one edge-cut of G all of whose edges receive different colours. In this note it is proved that if G has at least 2 vertices and has no bridges, then h(G) = |E(G)| −|V(G)|+2.

Keywords: anti-Ramsey, totally multicoloured, edge-cuts.

2000 Mathematics Subject Classification: 05C15, 05C40.

References

[1] N. Alon, On a Conjecture of Erdös, Simonovits and Sós Concerning Anti-Ramsey Theorems, J. Graph Theory 7 (1983) 91-94, doi: 10.1002/jgt.3190070112.
[2] P. Erdös, M. Simonovits and V.T. Sós, Anti-Ramsey Theorems, in: Infinite and finite sets (Keszthely, Hungary 1973), Colloquia Mathematica Societatis János Bolyai, 10, (North-Holland, Amsterdam, 1975) 633-643.
[3] P. Hell and J.J. Montellano-Ballesteros, Polychromatic Cliques, Discrete Math. 285 (2004) 319-322, doi: 10.1016/j.disc.2004.02.013.
[4] T. Jiang, Edge-colorings with no Large Polychromatic Stars, Graphs and Combinatorics 18 (2002) 303-308, doi: 10.1007/s003730200022.
[5] J.J. Montellano-Ballesteros, On Totally Multicolored Stars, to appear J. Graph Theory.
[6] J.J. Montellano-Ballesteros and V. Neumann-Lara, An Anti-Ramsey Theorem, Combinatorica 22 (2002) 445-449, doi: 10.1007/s004930200023.
[7] M. Simonovits and V.T. Sós, On Restricted Colourings of Kn, Combinatorica 4 (1984) 101-110, doi: 10.1007/BF02579162.
[8] H. Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932) 339-362, doi: 10.1090/S0002-9947-1932-1501641-2.

Received 18 September 2004
Revised 28 November 2005