On the number of edges in maximally linkless graphs

On the Average Number of Edges in Theta Graphs

Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimumspanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds—with vanishing error terms—hold for theta gr...

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [2] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for seve...

On the extremal number of edges in hamiltonian connected graphs

Assume that n and δ are positive integers with 3 ≤ δ < n. Let hc(n, δ) be the minimum number of edges required to guarantee ann-vertex graphGwithminimumdegree δ(G) ≥ δ to be hamiltonian connected. Any n-vertex graphGwith δ(G) ≥ δ is hamiltonian connected if |E(G)| ≥ hc(n, δ). We prove that hc(n, δ) = C(n − δ + 1, 2) + δ2 − δ + 1 if δ ≤ b n+3×(n mod 2) 6 c+1,hc(n, δ) = C(n−b n 2 c+1, 2)+b n 2 c ...

On the Extremal Number of Edges in Hamiltonian Graphs

TUNG-YANG HO, CHENG-KUAN LIN, JIMMY J. M. TAN, D. FRANK HSU AND LIH-HSING HSU Department of Information Management Ta Hwa Institute of Technology Hsinchu, 307 Taiwan E-mail: [email protected] Department of Computer Science National Chiao Tung University Hsinchu, 300 Taiwan Department of Computer and Information Science Fordham University New York, NY 10023, U.S.A. Department of Computer Science ...

On the Number of Edges in Random Planar Graphs