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An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $vec{chi}(G)$, is the minimum maximum in-degree of a proper orientation of G. Araujo et al. (Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant $c$ such that $vec{chi}(G)leq c$ for every outerplanar graph $G$ and showed that $vec{chi}(G)leq 7$ for every cactus $G.$ We prove that $vec{chi}(G)leq 3$ if $G$ is a triangle-free $2$-connected outerplanar graph and $vec{chi}(G)leq 4$ if $G$ is a triangle-free bridgeless outerplanar graph.
A semi-proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: A(D)rightarrow mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denote
For all $nge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by ErdH{o}s, and extends results by Grzesik and Hatami, Hladky, Kr{
Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one tri
In this paper, we investigate the ratio of the numbers of odd and even cycles in outerplanar graphs. We verify that the ratio generally diverges to infinity as the order of a graph diverges to infinity. We also give sharp estimations of the ratio for
In this short note, we show that for any $epsilon >0$ and $k<n^{0.5-epsilon}$ the choice number of the Kneser graph $KG_{n,k}$ is $Theta (nlog n)$.