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- W2811228059 abstract "Abstract An edge-coloured graph G is called properly connected if every two vertices are connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. van Aardt et al. (2017)gave a sufficient condition for the proper connection number to be at most k in terms of the size of graphs. In this note, our main result is the following, by adding a minimum degree condition: let G be a connected graph of order n, k ≥ 3. If | E ( G ) | ≥ ( n − m − ( k + 1 − m ) ( δ + 1 ) 2 ) + ( k + 1 − m ) ( δ + 1 2 ) + k + 2 , then pc(G) ≤ k, where m takes the value k + 1 if δ = 1 and ⌊ k δ − 1 ⌋ if δ ≥ 2. Furthermore, if k = 2 and δ = 2 , pc(G) ≤ 2, except G ∈ {G1, Gn} (n ≥ 8), where G 1 = K 1 ∨ 3 K 2 and Gn is obtained by taking a complete graph K n − 5 and K1∨(2K2) with an arbitrary vertex of K n − 5 and a vertex with d ( v ) = 4 in K1∨(2K2) being joined. If k = 2 , δ ≥ 3, we conjecture pc(G) ≤ 2, where m takes the value 1 if δ = 3 and 0 if δ ≥ 4 in the assumption." @default.
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- W2811228059 date "2019-07-01" @default.
- W2811228059 modified "2023-09-24" @default.
- W2811228059 title "Minimum degree and size conditions for the proper connection number of graphs" @default.
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- W2811228059 doi "https://doi.org/10.1016/j.amc.2019.01.062" @default.
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