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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 densities in arbitrary large graphs. Huang, Linial, Naves, Peled and Sudakov have recently determined the projection of the set S to the (d_0,d_3) plane. We determine the projection of the set S to all the remaining planes.
A path in a vertex-colored graph is called emph{conflict free} if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be emph{conflict-free vertex-connected} if any two vertices of the graph are connected by a conf
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
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. A
A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $omegacolon V(G)tomathbb{R}$ and any list assignment $Lcolon E(G)to2^{mathbb{R}}$ with $|L(e)|geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)in L(e)$ for all $ei
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,varph