ﻻ يوجد ملخص باللغة العربية
A (vertex) $ell$-ranking is a labelling $varphi:V(G)tomathbb{N}$ of the vertices of a graph $G$ with integer colours so that for any path $u_0,ldots,u_p$ of length at most $ell$, $varphi(u_0) eqvarphi(u_p)$ or $varphi(u_0)<max{varphi(u_0),ldots,varphi(u_p)}$. We show that, for any fixed integer $ellge 2$, every $n$-vertex planar graph has an $ell$-ranking using $O(log n/logloglog n)$ colours and this is tight even when $ell=2$; for infinitely many values of $n$, there are $n$-vertex planar graphs, for which any 2-ranking requires $Omega(log n/logloglog n)$ colours. This result also extends to bounded genus graphs. In developing this proof we obtain optimal bounds on the number of colours needed for $ell$-ranking graphs of treewidth $t$ and graphs of simple treewidth $t$. These upper bounds are constructive and give $O(nlog n)$-time algorithms. Additional results that come from our techniques include new sublogarithmic upper bounds on the number of colours needed for $ell$-rankings of apex minor-free graphs and $k$-planar graphs.
We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the follo
Given a graph $G=(V,E)$ and a positive integer $tgeq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $Fsubseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VC
For a planar graph with a given f-vector $(f_{0}, f_{1}, f_{2}),$ we introduce a cubic polynomial whose coefficients depend on the f-vector. The planar graph is said to be real if all the roots of the corresponding polynomial are real. Thus we have a
Let d_i(G) be the density of the 3-vertex i-edge graph in a graph G, i.e., the probability that three random vertices induce a subgraph with i edges. Let S be the set of all quadruples (d_0,d_1,d_2,d_3) that are arbitrary close to 3-vertex graph dens
Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a emph{distributed interactive proof} for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed