Oriented diameter of graphs with given maximum degree

Abstract

In this article, we show that every bridgeless graph G of order n and maximum degree Δ has an orientation of diameter at most n−Δ+3. We then use this result and the definition NGfalse(Hfalse)=⋃v∈V(H)NGfalse(vfalse)∖Vfalse(Hfalse), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high‐degree vertices, namely: For every edge e, G has an orientation of diameter at most n−false|NG(e)false|+4, if e is on a triangle and at most n−false|NG(e)false|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n−false|NG(H)false|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|− deg Gfalse(sfalse))+7, for every partite set A of G and s∈V(G)∖A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n−2Δ+7. For each bound, we give a polynomial‐time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp.

Diameter Three Orientability of Bipartite Graphs

“…Since then, several upper bounds on − −− → diam(G) in terms of certain graph invariants such as the domination number [8,11], the maximum degree [7] and the minimum degree [2,18] have been given. For a survey on the numerous further results on the oriented diameter published over the past decades, see, e.g., Koh and Tay [10].…”

“…In 2018, Dankelmann, Guo and Surmacs [7] considered the oriented diameter of a bridgeless graph G of order n in terms of the maximum degree. They proved the upper bound − −− → diam(G) n − ∆(G) + 3.…”

“…Furthermore, they showed that, if G is bipartite, the bound above can be improved: Theorem 3. (Dankelmann, Guo and Surmacs [7]) Let G be a bridgeless bipartite graph with partite sets X and Y , v ∈ Y . Then,…”

“…For a bridgeless balanced bipartite graph, that is, a bipartite graph whose partite sets have the same cardinality, by Theorem 3, a straightforward result is the following: Corollary 4. [7] Let G be a bridgeless balanced bipartite graph of order n and maximum degree ∆(G). Then,…”

Oriented diameter of maximal outerplanar graphs

“…Cochran et al [4] determined the minimum size of graphs of order that guarantees that its oriented diameter equals 2. Dankelmann et al [6] and Surmacs [13] obtained the following upper bounds for the oriented diameter in terms of the maximum or minimum degree, respectively. The latter result improved on a similar bound by Bau and Dankelmann [2].…”