Abstract
A Nordhaus–Gaddum-type result is a (tight) lower or upper bound on the sum (or product) of a parameter of a graph and its complement. The path covering number \(c(G)\) of a graph is the smallest number of vertex-disjoint paths needed to cover the vertices of the graph. For two positive integers \(j\) and \(k\) with \(j\ge k,\) an \(L(j,k)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(j,\) and the difference between labels of vertices that are distance two apart is at least \(k.\) The span of an \(L(j,k)\)-labeling of a graph \(G\) is the difference between the maximum and minimum integers used by it. The \(L(j,k)\)-labelings-number of \(G\) is the minimum span over all \(L(j,k)\)-labelings of \(G.\) This paper focuses on Nordhaus–Gaddum-type results for path covering and \(L(2,1)\)-labeling numbers.
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References
Balakrishnan H, Deo N (2003) Parallel algorithm for radiocoloring a graph. Congr Numer 160:193–204
Calamoneri T (2011) The \(L(h, k)\)-labelling problem: an updated survey and annotated bibliography. Comput J 54(8):1344–1371
Calamoneri T, Petreschi R (2004) \(L(h,1)\)-Labeling subclasses of planar graphs. J Parallel Distrib Comput 64(3):414–426
Calamoneri T, Pelc A, Petreschi R (2006) Labeling trees with a condition at distance two. Discret Math 306(14):1534–1539
Chang GJ, Kuo D (1996) The \(L(2,1)\)-labelling problem on graphs. SIAM J Discret Math 9:309–316
Georges JP, Mauro DW (1995) Generalized vertex labelings with a condition at distance two. Congr Numer 109:141–159
Georges JP, Mauro DW (1999) Some results on \(\lambda _k^j\)-numbers of the products of complete graphs. Congr Numer 140:141–160
Georges JP, Mauro DW, Whittlesey MA (1994) Relating path coverings to vertex labellings with a condition at distance two. Discret Math 135:103–111
Georges JP, Mauro DW, Stein MI (2000) Labeling products of complete graphs with a condition at distance two. SIAM J Discret Math 14:28–35
Griggs JR, Yeh RK (1992) Labelling graphs with a condition at distance 2. SIAM J Discret Math 5:586–595
Hong Y, Shu JL (2000) A sharp upper bound for the spectral radius of the Nordhaus–Gaddum type. Discret Math 211:229–232
Jha PK, Narayanan A, Sood P, Sundaram K, Sunder V (2000) On \(L(2,1)\)-labelling of the Cartesian product of a cycle and a path. Ars Comb 55:81–89
Li XL (1996) The relations between the spectral radius of a graph and its complement. J North China Technol Inst 17(4):297–299
Lü DM, Lü JJ (2005) A lower bound and an upper bound for the algebraic connectivity of Nordhaus–Gaddum type for similarity double-stars. J Nantong Univ (Nat Sci) 4(1):22
Lü DM, Lü JJ (2006) A bound for the algebraic connectivity of Nordhaus–Gaddum type for unicyclic graphs. J Zhejiang Univ (Sci Ed) 33(4):368–371
Lü D, Lin W, Song Z (2007) On \(L(2,1)\)-labelings of graphs with conectiving number \(k\). J Jilin Univ (Sci Ed) 45:555–561
Nordhaus EA, Gaddum JW (1956) On complementary graphs. Am Math Mon 63:175–177
Nosal E (1970) Eigenvalues of graphs. Masters’s Thesis, University of Calgary, Calgary
Sakai D (1994) Labelling chordal graphs: distance two condition. SIAM J Discret Math 7:133–140
Shu JL, Hong Y (2000) New upper bounds on sum of the spectral radius of a graph and its complement. J East China Norm Univ (Nat Sci) 6:13–17
Whittlesey MA, Georges JP, Mauro DW (1995) On the \(\lambda \)-number of \(Q_n\) and related graphs. SIAM J Discret Math 8:499–506
Zhou B (1997) A note about the relations between the spectral radius of a graph and its complement. Pure Appl Math 1391:15–18
Acknowledgments
This study was supported by NSFC, JSSPITP 2012JSSPITP1555, and the Natural Science Foundation of Nantong University 11Z055, 11Z056 and 11Z059.
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Lü, D., Du, J., Lin, N. et al. Nordhaus–Gaddum-type results for path covering and \(L(2,1)\)-labeling numbers. J Comb Optim 29, 502–510 (2015). https://doi.org/10.1007/s10878-013-9610-3
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DOI: https://doi.org/10.1007/s10878-013-9610-3