{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T03:06:46Z","timestamp":1787368006620,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>After a sequence of improvements Boyd et al. [TSP on cubic and subcubic graphs, Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6655, Springer, Heidelberg, 2011, pp. 65--77] proved that any 2-connected graph whose $n$ vertices have degree 3, i.e., a cubic 2-connected graph, has a Hamiltonian tour of length at most (4\/3)n, establishing in particular that the integrality gap of the subtour LP is at most 4\/3 for cubic 2-connected graphs and matching the conjectured value of the famous 4\/3 conjecture. In this paper we improve upon this result by designing an algorithm that finds a tour of length (4\/3-1\/61236)n, implying that cubic 2-connected graphs are among the few interesting classes of graphs for which the integrality gap of the subtour LP is strictly less than 4\/3. With the previous result, and by considering an even smaller $\\epsilon$, we show that the integrality gap of the TSP relaxation is at most $4\/3- \\epsilon$ even if the graph is not 2-connected (i.e., for cubic connected graphs), implying that the approximability threshold of the TSP in cubic graphs is strictly below 4\/3. Finally, using similar techniques we show, as an additional result, that every Barnette graph admits a tour of length at most (4\/3 - 1\/18)n.<\/jats:p>","DOI":"10.1137\/140972925","type":"journal-article","created":{"date-parts":[[2015,5,21]],"date-time":"2015-05-21T11:19:29Z","timestamp":1432207169000},"page":"915-939","source":"Crossref","is-referenced-by-count":17,"title":["TSP Tours in Cubic Graphs: Beyond 4\/3"],"prefix":"10.1137","volume":"29","author":[{"given":"Jos\u00e9","family":"Correa","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Larr\u00e9","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jos\u00e9 A.","family":"Soto","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,5,21]]},"reference":[{"key":"atypb1","unstructured":"N. Aggarwal, N. Garg, and S. Gupta,\n                      A 4\/3-approximation for TSP on cubic 3-edge-connected graphs\n                      , arXiv:1101.5586v1, 2011."},{"key":"atypb2","first-page":"875","author":"An H.-C.","year":"2012","journal-title":"New York"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480102409619"},{"key":"atypb4","unstructured":"D. Barnette,\n                      Conjecture\n                      5, in Recent Progress in Combinatorics: Proceedings of the Third Waterloo Conference on Combinatorics, W. T. Tutte, ed., Academic Press, New York, 1968."},{"key":"atypb5","first-page":"65","author":"Boyd S.","year":"2011","journal-title":"Heidelberg"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"J. A. Bondy and U. S. R. Murty,\n                      Graph Theory With Applications\n                      , Elsevier, New York, 1976.","DOI":"10.1007\/978-1-349-03521-2"},{"key":"atypb7","unstructured":"N. Christofides,\n                      Worst-Case Analysis of a New Heuristic for The Travelling Salesman Problem\n                      , Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA, 1976."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.orl.2004.09.005"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585563"},{"key":"atypb10","unstructured":"J. A. Karp and R. Ravi,\n                      A 9\/7-approximation algorithm of graphic TSP in cubic bipartite graphs\n                      , arXiv:1311.3640, 2013."},{"key":"atypb11","unstructured":"M. Karpinski and R. Schmied,\n                      Approximation hardness of graphic TSP on cubic graphs\n                      , arXiv:1304.6800v2, 2013."},{"key":"atypb12","unstructured":"O. A. Larre\u0301,\n                      El Problema del Vendedor Viajero en Grafos Cu\u0301bicos\n                      , Master's thesis, Universidad de Chile, Santiago, Chile, 2012."},{"key":"atypb13","first-page":"243","author":"Lampis M.","year":"2012","journal-title":"Heidelberg"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"T. Mo\u0308mke and O. Svensson,\n                      Approximating graphic TSP by matchings\n                      , in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science-FOCS 2011, IEEE Computer Society, Los Alamitos, CA, 2011, pp. 560-569.","DOI":"10.1109\/FOCS.2011.56"},{"key":"atypb15","unstructured":"M. Mucha,\n                      13\/9-approximation for graphic TSP\n                      , in Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science, LIPIcs. Leibniz Int. Proc. Inform. 14, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2012, pp. 30-41."},{"key":"atypb16","first-page":"550","author":"Gharan S. Oveis","year":"2011","journal-title":"Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-006-0008-z"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-014-0835-4"},{"key":"atypb19","first-page":"1477","author":"Schalekamp F.","year":"2012","journal-title":"New York"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"A. Sebo\u0308,\n                      Eight-Fifth Approximation for TSP Paths\n                      , Proceedings of the 16th Conference on Integer Programming and Combinatorial Optimization (IPCO), Lecture Notes in Comput. Sci. 7801, Springer, Heidelberg, 2013, pp. 362-374.","DOI":"10.1007\/978-3-642-36694-9_31"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-014-2960-3"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1080\/14786448408627475"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.26493\/1855-3974.193.8e7"},{"key":"atypb24","first-page":"169","volume":"7","author":"Tutte W. T.","year":"1946","journal-title":"J. London Math. Soc."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.2307\/2371125"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0120913"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/140972925","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:25:06Z","timestamp":1787322306000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/140972925"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/140972925"],"URL":"https:\/\/doi.org\/10.1137\/140972925","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}