The path covering number (or path-covering number; Slater 1972) of a graph , variously denoted as summarized below, is the minimum number
of vertex-disjoint paths that cover the vertices of .
notation
reference
Boesch et al. (1974)
Slater (1979)
DeLa Vina and Waller (2002)
Goedgebeur
et al. (2019)
Lu and Zhou (2013)
In order for the path covering number to be well-defined (for example, in the case of the claw graph, for which one or two vertices are "left over"
after covering with paths of length 2 or 1, respectively), "paths" consisting
of a single point must be allowed (Boesch et al. 1974).
A graph therefore has path covering number 1 iff it is traceable (Boesch et al. 1974).
A disconnected graph has a path covering number equal to the sum of the path covering numbers of its connected components.
Boesch (et al. 1974) gives values for a number of parametrized classes of
graphs.
Boesch, F. T.; Chen, S.; and McHugh, J. A. M. "On Covering the Points of a Graph with Point Disjoint Paths." In Graphs
and Combinatorics (Ed. R. A. Bari and F. Harary). Berlin: Springer-Verlag,
pp. 201-212, 1974.DeLa Vina, E. and Waller, B. "Independence,
Radius and Path Coverings in Trees." Congr. Numer.156, 155-169,
2002.Goedgebeur, J.; Ozeki, K.; van Cleemput, N.; and Wiener, G. "On
the Minimum Leaf Number of Cubic Graphs." Disc. Math.342, 3000-3005,
2019.Lovasz, L. Combinatorial Problems and Exercises. Academiai
Kiado, 1979.Lu, C. and Zhou, Q. "Path Covering Number and -Labeling Number of Graphs."
Disc. Appl. Math.161, 2062-2074, 2013.Ore, Ø.
"Arc Coverings of Graphs." Ann. Mat. Pura Appl.55, 315-332,
1961.Slater, P. J. "Path Coverings of the Vertices of a Tree."
Disc. Math.25, 65-74, 1979.