Algorithms and Almost Tight Results for 3-Colorability of Small Diameter Graphs [chapter]

George B. Mertzios, Paul G. Spirakis
2013 Lecture Notes in Computer Science  
Citation for published item: wertziosD qFfF nd pirkisD FqF @PHIQA 9elgorithms nd lmost tight results for QEolorility of smll dimeter grphsF9D in ypiw PHIQ X theory nd prtie of omputer siene X QWth interntionl onferene on urrent trends in theory nd prtie of omputer sieneD § pindler¦ uv wl¡ ynD gzeh epuliD tnury PTEQID PHIQF roeedingsF ferlinD reidelergX pringerD ppF QQPEQRQF veture notes in omputer sieneF @UURIAF Further information on publisher's website: Publisher's copyright statement: The
more » ... publication is available at Springer via http://dx.The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract. The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2 o(n) on graphs with n vertices and diameter at most 4. In spite of extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been an open problem. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2 O( √ n log n) . Furthermore we extend the notion of an articulation vertex to that of an articulation neighborhood, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2 that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish the complexity of 3-coloring by proving for every ε ∈ [0, 1) that 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree δ = Θ(n ε ). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every ε ∈ [0, 1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree δ = Θ(n ε ). Finally, we provide a 3-coloring algorithm with running time 2 O(min{δ∆, n δ log δ}) for arbitrary graphs with diameter 3, where n is the number of vertices and δ (resp. ∆) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this is the first subexponential algorithm for graphs with δ = ω(1) and for graphs with δ = O(1) and ∆ = o(n). Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree of the input graph is δ = Θ(n ε ), where ε ∈ [ 1 2 , 1), as its time complexity is 2 O( n δ log δ) = 2 O(n 1−ε log n) and the corresponding lower bound states that there is no 2 o(n 1−ε ) -time algorithm.
doi:10.1007/978-3-642-35843-2_29 fatcat:3segyayej5azzpyhuu2eqfpjoa