Long Paths and Hamiltonian paths in Inhomogenous Random Graphs

In this paper, we study long paths and Hamiltonian paths in inhomogenous random graphs. In the first part of the paper, we consider an inhomogenous Erd\H{o}s-R\'enyi random graph $G_E$ with average edge density $p_n.$ We prove that if $np_n^2 \longrightarrow \infty$ as $n \rightarrow \infty,$ then the longest path contains at least $n-ne^{-\delta_1 np_n^2}$ nodes with high probability (i.e., with probability converging to one as $n \rightarrow \infty$), for some constant $\delta_1> 0 .$ In particular, if $np_n^2 = M\log{n}$ for some constant $M > 0$ large, then $G_E$ is Hamiltonian with high probability; i.e., the longest path contains all the nodes of $G_E.$ In the second part of the paper, we consider a random geometric graph $G_R$ consisting of $n$ nodes, each independently distributed according to a (not necessarily uniform) density $f.$ If $r_n$ is the connectivity radius and $nr_n^2 \longrightarrow \infty,$ then with high probability, the longest cycle contains at least $n-ne^{-\delta_2 nr_n^2}$ nodes for some constant $\delta_2 > 0.$ As a consequence of our proof, we obtain that if $nr_n^2 = \log{n} + 7\log{\log{n}} + \omega_n$ and $\omega_n \longrightarrow \infty$ as $n \rightarrow \infty,$ then with high probability $G_R$ contains a Hamiltonian cycle.