On 3-Coloring Circle Graphs
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10493555" target="_blank" >RIV/00216208:11320/24:10493555 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=agQuh34HRK" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=agQuh34HRK</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.7155/jgaa.v28i1.2991" target="_blank" >10.7155/jgaa.v28i1.2991</a>
Alternative languages
Result language
angličtina
Original language name
On 3-Coloring Circle Graphs
Original language description
Given a graph G with a fixed vertex order PRECEDES, one obtains a circle graph H whose vertices are the edges of G and where two such edges are adjacent if and only if their endpoints are pairwise distinct and alternate in PRECEDES. Therefore, the problem of determining whether G has a k-page book embedding with spine order PRECEDES is equivalent to deciding whether H can be colored with k colors. Finding a k-coloring for a circle graph is known to be NP-complete for k >= 4 [9] and trivial for k <= 2. For k = 3, Unger (1992) claims an efficient algorithm that finds a 3-coloring in O(n log n) time, if it exists. Given a circle graph H, Unger's algorithm (1) constructs a 3-Sat formula Φ that is satisfiable if and only if H admits a 3-coloring and (2) solves Φ by a backtracking strategy that relies on the structure imposed by the circle graph. However, the extended abstract misses several details and Unger refers to his PhD thesis (in German) for details. In this paper we argue that Unger's algorithm for 3-coloring circle graphs is not correct and that 3-coloring circle graphs should be considered as an open problem. We show that step (1) of Unger's algorithm is incorrect by exhibiting a circle graph and its representation whose formula Φ is satisfiable but that is not 3-colorable. We further show that Unger's backtracking strategy for solving Φ in step (2) may produce incorrect results and give empirical evidence that it exhibits a runtime behaviour that is not consistent with the claimed running time.
Czech name
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Czech description
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Classification
Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Graph Algorithms and Applications
ISSN
1526-1719
e-ISSN
1526-1719
Volume of the periodical
28
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
14
Pages from-to
389-402
UT code for WoS article
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EID of the result in the Scopus database
2-s2.0-85209207431