Further results on the total monochromatic connectivity of graphs

Yanhong Gao (), Ping Li () and Xueliang Li ()
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Yanhong Gao: Nankai University
Ping Li: Nankai University
Xueliang Li: Nankai University

Journal of Combinatorial Optimization, 2022, vol. 44, issue 1, No 29, 603-616

Abstract: Abstract The concepts of monochromatic connection number mc(G) (MC-number for short) and vertex monochromatic connection number mvc(G) (MVC-number for short) of a graph G were introduced in 2011 and 2018, respectively, by Caro and Yuster and Cai et al., and have been studied extensively, While in 2017, Jiang et al. introduced the concept of total monochromatic connection number tmc(G) (TMC-number for shot) of a graph G. In this paper, we mainly study the TMC-number of a graph. At first, we completely determine the TMC-numbers for any given simple and connected graphs, and obtain some Nordhaus-Gaddum-type results for the TMC-number. Jiang et al. in 2017 put forward a conjecture and a problem on the difference between tmc(G), mc(G) and mvc(G) of a graph G. We then completely solve the conjecture and the problem, and characterize the graphs G of order n with $$tmc(G)-mc(G)=n-1$$ t m c ( G ) - m c ( G ) = n - 1 .

Keywords: Total monochromatic connection coloring (number); Nordhaus-Gaddum-type result; Edge (vertex) monochromatic connection number (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-022-00850-5

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