Iterated arc graphs

Danny Rorabaugh; Claude Tardif; David Wehlau; Imed Zaguia

Commentationes Mathematicae Universitatis Carolinae (2018)

  • Volume: 59, Issue: 3, page 277-283
  • ISSN: 0010-2628

Abstract

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The arc graph δ ( G ) of a digraph G is the digraph with the set of arcs of G as vertex-set, where the arcs of δ ( G ) join consecutive arcs of G . In 1981, S. Poljak and V. Rödl characterized the chromatic number of δ ( G ) in terms of the chromatic number of G when G is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph δ k ( G ) still only depends on the chromatic number of G when G is symmetric. Our proof is a rediscovery of the proof of [Poljak S., Coloring digraphs by iterated antichains, Comment. Math. Univ. Carolin. 32 (1991), no. 2, 209-212], though various mistakes make the original proof unreadable.

How to cite

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Rorabaugh, Danny, et al. "Iterated arc graphs." Commentationes Mathematicae Universitatis Carolinae 59.3 (2018): 277-283. <http://eudml.org/doc/294387>.

@article{Rorabaugh2018,
abstract = {The arc graph $\delta (G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $\delta (G)$ join consecutive arcs of $G$. In 1981, S. Poljak and V. Rödl characterized the chromatic number of $\delta (G)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph $\delta ^k(G)$ still only depends on the chromatic number of $G$ when $G$ is symmetric. Our proof is a rediscovery of the proof of [Poljak S., Coloring digraphs by iterated antichains, Comment. Math. Univ. Carolin. 32 (1991), no. 2, 209-212], though various mistakes make the original proof unreadable.},
author = {Rorabaugh, Danny, Tardif, Claude, Wehlau, David, Zaguia, Imed},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {arc graph; chromatic number; free distributive lattice; Dedekind number},
language = {eng},
number = {3},
pages = {277-283},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Iterated arc graphs},
url = {http://eudml.org/doc/294387},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Rorabaugh, Danny
AU - Tardif, Claude
AU - Wehlau, David
AU - Zaguia, Imed
TI - Iterated arc graphs
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 3
SP - 277
EP - 283
AB - The arc graph $\delta (G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $\delta (G)$ join consecutive arcs of $G$. In 1981, S. Poljak and V. Rödl characterized the chromatic number of $\delta (G)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph $\delta ^k(G)$ still only depends on the chromatic number of $G$ when $G$ is symmetric. Our proof is a rediscovery of the proof of [Poljak S., Coloring digraphs by iterated antichains, Comment. Math. Univ. Carolin. 32 (1991), no. 2, 209-212], though various mistakes make the original proof unreadable.
LA - eng
KW - arc graph; chromatic number; free distributive lattice; Dedekind number
UR - http://eudml.org/doc/294387
ER -

References

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  5. Markowsky G., 10.1016/0012-365X(80)90156-9, Discrete Math. 29 (1980), no. 3, 275–285. MR0560771DOI10.1016/0012-365X(80)90156-9
  6. McHard R. W., Sperner Properties of the Ideals of a Boolean Lattice, Ph.D. Thesis, University of California, Riverside, 2009. MR2718021
  7. Poljak S., Coloring digraphs by iterated antichains, Comment. Math. Univ. Carolin. 32 (1991), no. 2, 209–212. MR1137780
  8. Poljak S., Rödl V., 10.1016/S0095-8956(81)80024-X, J. Combin. Theory Ser. B 31 (1981), no. 2, 190–198. MR0630982DOI10.1016/S0095-8956(81)80024-X

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