Abstract
Graph coloring has interesting applications in optimization, calculation of Hessian matrix, network design and so on. In this paper, we consider an improper edge coloring which is one important coloring-linear arboricity. For a graph G, a linear forest is a disjoint union of paths and cycles. The linear arboricity la(G) is the minimum number of disjoint linear forests such that their union is exactly the edge set of G. In this paper, we study a special case that G is a simple planar graph with two not adjacent cycles each with a chordal and length between 4 and 7. We show that in this special case, \(la(G)=\lceil \frac{\Delta }{2}\rceil \) where \(\Delta \) is the maximum vertex degree of G and \(\Delta \ge 7\).
Similar content being viewed by others
References
Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs III: cyclic and acyclic invariants. Math. Slovaca 30, 405–417 (1980)
Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs IV: linear arboricity. Networks 11, 69–72 (1981)
Alon, N.: The linear arboricity of graphs. Isr. J. Math. 62, 311–325 (1988)
Alon, N., Teague, V.J., wormald, N.C.: Linear arboricity and linear \(k\)-arboricity of regular graphs. Graphs Comb. 17, 11–16 (2001)
Angelini, P., Frati, F.: Acyclically 3-colorable planar graphs. J Comb. Optim. 24, 116–130 (2012)
Bessy, S., Havet, F.: Enumerating the edge-colourings and total colourings of a regular graph. J Comb. Optim. 25, 523–535 (2013)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, New York (1976)
Cygan, M., Hou, J.F., Kowalik, L., Luzar, B., Wu, J.L.: A planar linear arboricity conjecture. J. Graph Theory 69, 403–425 (2012)
Du, H.W., Jia, X.H., Li, D.Y., Wu, W.L.: Coloring of double disk graphs. J. Global Optim. 28, 115–119 (2004)
Li, X.W., Mak-Hau, V., Zhou, S.M.: The \(L(2,1)\)-labelling problem for cubic Cayley graphs on dihedral groups. J Comb. Optim. 25, 716–736 (2013)
Li, Y., Hu, X.: The linear 2-arboricity of sparse graphs. Discrete Math. Algorithms Appl. 9, 1750047 (2017)
Orponen, P., Russo, D.A.: Uwe Schōning: optimal approximations and polynomially levelable sets. SIAM J. Comput. 15(2), 399–408 (1986)
Péoche, B.: Complexity of the linear arboricity of a graph. RAIRO Rech. Oper. 16, 125–129 (1982)
Wang, H.J., Liu, B., Gu, Y., Zhang, X., Wu, W.L., Gao, H.W.: Total coloring of planar graphs without adjacent short cycles. J. Comb. Optim. 33, 265–274 (2017)
Wang, H.J., Wu, J.L., Liu, B., Chen, H.Y.: On the linear arboricity of graphs embeddable in surfaces. Inf. Process. Lett. 114, 475–479 (2014)
Wang, H.J., Wu, L.D., Wu, W.L., Pardalos, P.M., Wu, J.L.: Minimum total coloring of planar graph. J. Global Optim. 60, 777–791 (2014)
Wang, H.J., Wu, L.D., Wu, W.L., Wu, J.L.: Minimum number of disjoint linear forests covering a planar graph. J. Comb. Optim. 28, 274–287 (2014)
Wang, H.J., Wu, L.D., Zhang, X., Wu, W.L., Liu, B.: A note on the minimum number of choosability of planar graphs. J. Comb. Optim. 31, 1013–1022 (2016)
Wu, J.L.: On the linear arboricity of planar graphs. J. Graph Theory 31, 129–134 (1999)
Wu, J.L., Wu, Y.W.: The linear arboricity of planar graphs of maximum degree seven are four. J. Graph Theory 58, 210–220 (2008)
Wu, J.L.: Some path decompositions of Halin graphs. J. Shandong Min. Inst. 15, 219–222 (1996)
Wu, J.L.: The linear arboricity of series-parallel graphs. Graphs Combin. 16, 367–372 (2000)
Wu, J.L., Hou, J.F., Sun, X.Y.: A note on the linear arboricity of planar graphs without 4-cycles. ISORA’09, pp. 174–178 (2009)
Acknowledgements
This work was supported in part by National Natural Science Foundation of China (Grant Nos. 11501316, 71171120, 71571180), China Postdoctoral Science Foundation (2016M600556), Qingdao Postdoctoral Application Research Project (2016156), Shandong Provincial Natural Science Foundation of China (Grant Nos. ZR2017QA010, ZR2017MF055).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, H., Wu, L., Pardalos, M.P. et al. An efficient case for computing minimum linear arboricity with small maximum degree. Optim Lett 13, 419–428 (2019). https://doi.org/10.1007/s11590-018-1278-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-018-1278-2