Nordhaus–Gaddum type result for the matching number of a graph

Huiqiu Lin (), Jinlong Shu and Baoyindureng Wu ()
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Huiqiu Lin: East China University of Science and Technology
Jinlong Shu: East China Normal University
Baoyindureng Wu: Xinjiang University

Journal of Combinatorial Optimization, 2017, vol. 34, issue 3, No 17, 916-930

Abstract: Abstract For a graph G, $$\alpha '(G)$$ α ′ ( G ) is the matching number of G. Let $$k\ge 2$$ k ≥ 2 be an integer, $$K_{n}$$ K n be the complete graph of order n. Assume that $$G_{1}, G_{2}, \ldots , G_{k}$$ G 1 , G 2 , … , G k is a k-decomposition of $$K_{n}$$ K n . In this paper, we show that (1) $$\begin{aligned} \left\lfloor \frac{n}{2}\right\rfloor \le \sum _{i=1}^{k} \alpha '(G_{i})\le k\left\lfloor \frac{n}{2}\right\rfloor . \end{aligned}$$ n 2 ≤ ∑ i = 1 k α ′ ( G i ) ≤ k n 2 . (2) If each $$G_{i}$$ G i is non-empty for $$i = 1, \ldots , k$$ i = 1 , … , k , then for $$n\ge 6k$$ n ≥ 6 k , $$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n+k-1}{2}\right\rfloor . \end{aligned}$$ ∑ i = 1 k α ′ ( G i ) ≥ n + k - 1 2 . (3) If $$G_{i}$$ G i has no isolated vertices for $$i = 1, \ldots , k$$ i = 1 , … , k , then for $$n\ge 8k$$ n ≥ 8 k , $$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n}{2}\right\rfloor +k. \end{aligned}$$ ∑ i = 1 k α ′ ( G i ) ≥ n 2 + k . The bounds in (1), (2) and (3) are sharp. (4) When $$k= 2$$ k = 2 , we characterize all the extremal graphs which attain the lower bounds in (1), (2) and (3), respectively.

Keywords: Decomposition; Matching number; Nordhaus–Gaddum type result; 05C50 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-017-0120-6

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