(Strong) Proper Connection in Some Digraphs

• Published in: IEEE access Vol. 7; pp. 69692 - 69697
• Main Authors: Ma, Yingbin, Nie, Kairui
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Apexes; Cacti; Color; Graph theory; Indexes; Information science; Mathematics; proper connection number; proper geodesic; Proper path; Rain; strong proper connection number; Terminology
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2019.2918368

An arc-colored digraph <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> is proper connected if any pair of vertices <inline-formula> <tex-math notation="LaTeX">v_{i},v_{j} \in V(D) </tex-math></inline-formula> there is a proper <inline-formula> <tex-math notation="LaTeX">v_{i}-v_{j} </tex-math></inline-formula> path whose adjacent arcs have different colors and a proper <inline-formula> <tex-math notation="LaTeX">v_{j}-v_{i} </tex-math></inline-formula> path whose adjacent arcs have different colors. The proper connection number of a digraph <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> is the minimum number of colors needed to make <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> proper connected, denoted by <inline-formula> <tex-math notation="LaTeX">\overrightarrow {pc}(D) </tex-math></inline-formula>. An arc-colored digraph <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> is strong proper connected if any pair of vertices <inline-formula> <tex-math notation="LaTeX">v_{i},v_{j} \in V(D) </tex-math></inline-formula> there is a proper <inline-formula> <tex-math notation="LaTeX">v_{i}-v_{j} </tex-math></inline-formula> geodesic and a proper <inline-formula> <tex-math notation="LaTeX">v_{j}-v_{i} </tex-math></inline-formula> geodesic. The strong proper connection number of <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> is the minimum number of colors required to color the arcs of <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> in order to make <inline-formula> <tex-math notation="LaTeX">D </tex-math></inline-formula> strong proper connected, denoted by <inline-formula> <tex-math notation="LaTeX">\overrightarrow {spc}(D) </tex-math></inline-formula>. In this paper, we will show some results on <inline-formula> <tex-math notation="LaTeX">\overrightarrow {pc}(D) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\overrightarrow {spc}(D) </tex-math></inline-formula>, mostly for the case of the (strong) proper connection numbers of cacti and circulant digraphs.