We show that every complete n-vertex simple topological graph contains a subgraph on at least $$(\log n)^{1/4 - o(1)}$$ vertices is weakly isomorphic to the convex geometric or twisted graph. This first improvement bound $$\varOmega (\log ^{1/8}n)$$ obtained in 2003 by Pach, Solymosi, and Tóth. also plane path of length n)^{1 -o(1)}$$ .

Let G be a complete graph with n+ 1 vertices. In a recent paper of the authors, it is shown that the path trees of the graph play a special role in the structure of the truncated powers and partition functions that are associated with the graph. Motivated by the above, we take here a closer look at the geometry of the simplicial cones associated with the graph, and the role played by those simp...

The 2-Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing prespecified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer ` for which an input graph can be contracted to the path P` on ` vertices. We show that the ...

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC1, respectively. This solves an open problem from Balcázar, Gabarró, and Sántha. We also show that the simulation problem is P-complete even for graphs of bounded path-w...

We consider various edge disjoint partitions of complete bipartite graphs. One case is where we decompose the edge set into edge disjoint paths of increasing lengths. A graph G is path-perfect if there is a positive integer n such that the edge set E(G) of the graph G can be partitioned into paths of length 1, 2, 3, . . . , n. The main result of the paper is the proof of the conjecture of Fink ...

A squarefree word is a sequence w of symbols such that there are no strings x, y, and z for which w = xyyz. A nonrepetitive coloring of a graph is an edge coloring in which the sequence of colors along any open path is squarefree. We show that determining whether a graph G has a nonrepetitive k-coloring is Σ p 2 -complete. When we restrict to paths of lengths at most n, the problem becomes NP-c...

It is shown that the existence of an Euler path in a recursive graph is complete for the class DΣ 3 of all set differences of two Σ 3 sets. The same problem for highly recursive graphs as well as automatic graphs is shown to be Π 2 -complete. Moreover, the arithmetic level for bounding the number of ends in an automatic/recursive graph as well as computing the number of infinite paths in an aut...

Let G be a connected graph and Q(G) the signless Laplacian of G. The principal ratio γ(G) is maximum minimum entries Perron vector Q(G). In this paper, we consider among all graphs order n, show that for sufficiently large n extremal kite obtained by identifying an end vertex path to any complete graph.

let g be a connected simple (molecular) graph. the distance d(u, v) between two vertices u and v of g is equal to the length of a shortest path that connects u and v. in this paper we compute some distance based topological indices of h-phenylenic nanotorus. at first we obtain an exactformula for the wiener index. as application we calculate the schultz index and modified schultz index of this ...

We derive an exact closed-form analytical expression for the distribution of the cover time for a random walk over an arbitrary graph. In special case, we derive simplified exact expressions for the distributions of cover time for a complete graph, a cycle graph, and a path graph. An accurate approximation for the cover time distribution, with computational complexity of O(2n) , is also present...