Proper orientation, proper biorientation and semi-proper orientation numbers of graphs

J. Ai (), S. Gerke (), G. Gutin (), H. Lei () and Y. Shi ()
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J. Ai: Royal Holloway, University of London Egham
S. Gerke: Royal Holloway, University of London Egham
G. Gutin: Royal Holloway, University of London Egham
H. Lei: Nankai University
Y. Shi: Nankai University

Journal of Combinatorial Optimization, 2023, vol. 45, issue 1, No 42, 10 pages

Abstract: Abstract An orientation D of G is proper if for every $$xy\in E(G)$$ x y ∈ E ( G ) , we have $$d^-_D(x)\ne d^-_D(y)$$ d D - ( x ) ≠ d D - ( y ) . An orientation D is a p-orientation if the maximum in-degree of a vertex in D is at most p. The minimum integer p such that G has a proper p-orientation is called the proper orientation number pon(G) of G [introduced by Ahadi and Dehghan (Inf Process Lett 113:799–803, 2013)]. We introduce a proper biorientation of G, where an edge xy of G can be replaced by either arc xy or arc yx or both arcs xy and yx. Similarly to pon(G), we can define the proper biorientation number pbon(G) of G using biorientations instead of orientations. Clearly, $$\textrm{pbon}(G)\le \textrm{pon}(G)$$ pbon ( G ) ≤ pon ( G ) for every graph G. We compare pbon(G) with pon(G) for various classes of graphs. We show that for trees T, the tight bound $$\textrm{pon}(T)\le 4$$ pon ( T ) ≤ 4 extends to the tight bound $$\textrm{pbon}(T)\le 4$$ pbon ( T ) ≤ 4 and for cacti G, the tight bound $$\textrm{pon}(G)\le 7$$ pon ( G ) ≤ 7 extends to the tight bound $$\textrm{pbon}(G)\le 7.$$ pbon ( G ) ≤ 7 . We also prove that there is an infinite number of trees T for which $$\textrm{pbon}(T)

Keywords: Proper orientation; Biorientation; Semi-proper orientation; Trees; Cacti (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10878-022-00969-5

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