# On-line ranking number for cycles and paths

Discussiones Mathematicae Graph Theory (1999)

- Volume: 19, Issue: 2, page 175-197
- ISSN: 2083-5892

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topErik Bruoth, and Mirko Horňák. "On-line ranking number for cycles and paths." Discussiones Mathematicae Graph Theory 19.2 (1999): 175-197. <http://eudml.org/doc/270736>.

@article{ErikBruoth1999,

abstract = {A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number $χ*_r(G)$ of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that $χ*_r(Pₙ) < 3log₂n$ for n ≥ 2. Here we show that $χ*_r(Pₙ) ≤ 2⎣log₂n⎦+1$. The same upper bound is obtained for $χ*_r(Cₙ)$,n ≥ 3.},

author = {Erik Bruoth, Mirko Horňák},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {ranking number; on-line vertex colouring; cycle; path; colouring},

language = {eng},

number = {2},

pages = {175-197},

title = {On-line ranking number for cycles and paths},

url = {http://eudml.org/doc/270736},

volume = {19},

year = {1999},

}

TY - JOUR

AU - Erik Bruoth

AU - Mirko Horňák

TI - On-line ranking number for cycles and paths

JO - Discussiones Mathematicae Graph Theory

PY - 1999

VL - 19

IS - 2

SP - 175

EP - 197

AB - A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number $χ*_r(G)$ of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that $χ*_r(Pₙ) < 3log₂n$ for n ≥ 2. Here we show that $χ*_r(Pₙ) ≤ 2⎣log₂n⎦+1$. The same upper bound is obtained for $χ*_r(Cₙ)$,n ≥ 3.

LA - eng

KW - ranking number; on-line vertex colouring; cycle; path; colouring

UR - http://eudml.org/doc/270736

ER -

## References

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- [3] J.W.H. Liu, The role of elimination trees in sparse factorization, SIAM J. Matrix Analysis and Appl. 11 (1990) 134-172, doi: 10.1137/0611010. Zbl0697.65013
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- [5] I. Schiermeyer, Zs. Tuza and M. Voigt, On-line rankings of graphs, submitted.

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