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Minimum degree condition for proper connection number 2. (English) Zbl 1428.05109

Summary: A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph \(G\), the proper connection number \(\operatorname{pc}(G)\) of \(G\) is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of \(G\) is connected by at least one proper path in \(G\). Recently, X. Li and C. Magnant [“Properly colored notions of connectivity – a dynamic survey”, Theory Appl. Graphs Spec. Iss., Article 2 (2015; doi:10.20429/tag.2015.000102)] posed the following conjecture: If \(G\) is a connected noncomplete graph of order \(n \geq 5\) and minimum degree \(\delta(G) \geq n / 4\), then \(\operatorname{pc}(G) = 2\). In this paper, we show that this conjecture is true except for two small graphs on 7 and 8 vertices, respectively. As a byproduct we obtain that if \(G\) is a connected bipartite graph of order \(n \geq 4\) with \(\delta(G) \geq \frac{n + 6}{8}\), then \(\operatorname{pc}(G) = 2\).

MSC:

05C15 Coloring of graphs and hypergraphs
05C38 Paths and cycles
05C07 Vertex degrees
05C35 Extremal problems in graph theory
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References:

[1] Andrews, E.; Laforge, E.; Lumduanhom, C.; Zhang, P., On proper-path colorings in graphs, J. Combin. Math. Combin. Comput. (2016), to appear · Zbl 1347.05055
[2] Bondy, J. A.; Murty, U. S.R., Graph Theory, GTM, vol. 244 (2008), Springer · Zbl 1134.05001
[3] Borozan, V.; Fujita, S.; Gerek, A.; Magnant, C.; Manoussakis, Y.; Montero, L.; Tuza, Z., Proper connection of graphs, Discrete Math., 312, 2550-2560 (2012) · Zbl 1246.05090
[4] Chartrand, G.; Johns, G. L.; McKeon, K. A.; Zhang, P., Rainbow connection in graphs, Math. Bohem., 133, 85-98 (2008) · Zbl 1199.05106
[5] Gu, R.; Li, X.; Qin, Z., Proper connection number of random graphs, Theoret. Comput. Sci., 609, 336-343 (2016) · Zbl 1331.05194
[6] Huang, F.; Li, X.; Wang, S., Proper connection numbers of complementary graphs · Zbl 1393.05179
[7] Huang, F.; Li, X.; Wang, S., Proper connection number and 2-proper connection number of a graph
[8] Li, X.; Magnant, C., Properly colored notions of connectivity – a dynamic survey, Theory Appl. Graphs, 0, 1, Article 2 pp. (2015)
[9] Li, X.; Wei, M.; Yue, J., Proper connection number and connected dominating sets, Theoret. Comput. Sci., 607, 480-487 (2015) · Zbl 1333.05227
[10] Li, X.; Shi, Y.; Sun, Y., Rainbow connections of graphs: a survey, Graphs & Combin., 29, 1-38 (2013) · Zbl 1258.05058
[11] Li, X.; Sun, Y., Rainbow Connections of Graphs, Springer Briefs in Mathematics (2012), Springer: Springer New York · Zbl 1250.05066
[12] Williamsom, J. E., Panconnected graphs II, Period. Math. Hungar., 8, 2, 105-116 (1977) · Zbl 0339.05110
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