Longest and shortest cycles in random planar graphs

Abstract

Let P(n,m) be a graph chosen uniformly at random from the class of all planar graphs on vertex set {1,... ,n} with m = m(n) edges. We study the cycle and block structure of P(n,m) when m ∼ n/2. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in P(n,m) in the critical range when m = n/2 + o(n). In addition, we describe the block structure of P(n,m) in the weakly supercritical regime when(a) If s 3 n −2 → −∞, then whp L 1 is a tree and c (R) = Θ p n|s| −1 .

Concentration of maximum degree in random planar graphs

“…Together with (33) this implies (g), as K is strictly increasing. Similarly, we define for (h) the function g (x) := (x + 1/2) log x − x.…”

“…Kang, Moßhammer, and Sprüssel [34] showed that Theorem 2.10 is true for much more general classes of graphs. Prominent examples of such classes are cactus graphs, series-parallel graphs, and graphs embeddable on an orientable surface of genus g ∈ N ∪ {0} (see [33,Section 4]). Using the generalised version of Theorem 2.10 and analogous proofs of Theorems 1.4 and 1.6 and Corollaries 1.7 and 1.8, one can show the following.…”

Local limit of sparse random planar graphs

“…The only properties of P (n, m) used in our proofs are the statements on the internal structure from Theorem 2.5. Kang, Moßhammer, and Sprüssel [24] proved that Theorem 2.5 holds also for many other graph classes, including cactus graphs, series-parallel graphs, and graphs embeddable on an orientable surface of genus g ∈ N 0 (see also [23,Section 4]). Combining the generalised version of Theorem 2.5 and analogous proofs of Theorems 1.…”

Two point concentration of maximum degree in sparse random planar graphs

“…Kang, Moßhammer, and Sprüssel [34] showed that Theorem 2.9 is true for much more general classes of graphs. Prominent examples of such classes are cactus graphs, seriesparallel graphs, and graphs embeddable on an orientable surface of genus g ∈ N∪{0} (see [33,Section 4]). Using the generalised version of Theorem 2.9 and analogous proofs of Theorems 1.3 to 1.5 and Corollary 1.6, one can show the following.…”

“…Our proofs are based on the so-called core-kernel approach (see e.g. [5,[32][33][34][35]), which is a decomposition and construction technique for sparse graphs. We construct P stepwise and analyse each of the steps separately.…”