Abstract
Given a graph G, we study the problem of finding the minimum number of colors required for a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets consisting of colors of their incident edges. This minimum number is called the 2-distance vertex-distinguishing index, denoted by \(\chi '_{d2}(G)\). Using the breadth first search method, this paper provides a polynomial-time algorithm producing nearly-optimal solution in outerplanar graphs. More precisely, if G is an outerplanar graph with maximum degree \(\varDelta \), then the produced solution uses colors at most \(\varDelta +8\). Since \(\chi '_{d2}(G)\ge \varDelta \) for any graph G, our solution is within eight colors from optimal.
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Akbari, S., Bidkhori, H., Nosrati, N.: \(r\)-Strong edge colorings of graphs. Discrete Math. 306, 3005–3010 (2006)
Balister, P.N., Győri, E., Lehel, J., Schelp, R.H.: Adjacent vertex distinguishing edge-colorings. SIAM J. Discrete Math. 21, 237–250 (2007)
Bazgan, C., Harkat-Benhamdine, A.H., Li, H., Woźniak, M.: On the vertex-distinguishing proper edge-colorings of graphs. J. Comb. Theory Ser. B 75, 288–301 (1999)
Burris, A.C.: Vertex-distinguishingedge-colorings. Ph.D. Dissertation, Memphis State University (1993)
Burris, A.C., Schelp, R.H.: Vertex-distinguishing proper edge-colorings. J. Graph Theory 26, 73–82 (1997)
Calamoneri, T., Petreschi, R.: \(L(h,1)\)-labeling subclasses of planar graphs. J. Parallel. Distrib. Comput. 64, 414–426 (2004)
Chartrand, G., Harary, F.: Planar permutation graphs. Ann. Inst. H. Poincaŕe Sect. B (N.S.) 3, 433–438 (1967)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)
Hatami, H.: \(\varDelta +300\) is a bound on the the adjacent vertex distinguishing edge chromatic number. J. Comb. Theory Ser. B 95, 246–256 (2005)
Horňák, M., Huang, D., Wang, W.: On neighbor-distinguishing index of planar graphs. J. Graph Theory 76, 262–278 (2014)
Kemnitz, A., Marangio, M.: \(d\)-Strong edge colorings of graphs. Graphs Comb. 30, 183–195 (2014)
Mockovčiakvá, M., Soták, R.: Arbitrarily large difference between \(d\)-strong chromatic index and its trivial lower bound. Discrete Math. 313, 2000–2006 (2013)
Wang, W., Wang, Y., Huang, D., Wang, Y.: 2-Distance vertex-distinguishing edge coloring of graphs. Discrete Appl. Math. (Submitted) (2015)
Wang, W., Yue, X., Zhu, X.: The surviving rate of an outerplanar graph for the firefighter problem. Theor. Comput. Sci. 412, 913–921 (2011)
Wang, Y., Wang, W., Huo, J.: Some bounds on the neighbor-distinguishing index of graphs. Discrete Math. 338, 2006–2013 (2015)
Zhang, Z., Liu, L., Wang, J.: Adjacent strong edge coloring of graphs. Appl. Math. Lett. 15, 623–626 (2002)
Zhang, Z., Li, J., Chen, X., Cheng, H., Yao, B.: \(D(\beta )\)-vertex-distinguishing proper edge-coloring of graphs. Acta Math. Sinica (Chin. Ser.) 49, 703–708 (2006)
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Weifan Wang: Research supported by NSFC (No. 11371328).
Danjun Huang: Research supported by NSFC (Nos. 11301486, 11401535) and ZJNSFC (No. LQ13A010009).
Yiqiao Wang: Research supported by NSFC (No. 11301035).
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Wang, W., Huang, D., Wang, Y. et al. A polynomial-time nearly-optimal algorithm for an edge coloring problem in outerplanar graphs. J Glob Optim 65, 351–367 (2016). https://doi.org/10.1007/s10898-015-0360-x
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DOI: https://doi.org/10.1007/s10898-015-0360-x