Class Of 3-connected Graphs Research Articles

Enumeration of chordal planar graphs and maps

We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically g⋅n−5/2γnn! for a constant g>0 and γ≈11.89235. We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically s⋅n−3/2δn, where s>0, δ=1/σ≈6.40375, and σ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from K4 by repeatedly adding vertices adjacent to an existing triangular face.

Reduction for 3-Connected Graphs of Minimum Degree at Least Four

We give a reduction theorem for 3-connected graphs of minimum degree at least four. Let $${\mathcal{P}}_{3,4}$$ be the class of 3-connected graphs of minimum degree at least four. For a vertex x of degree four in G, splitting at x is an operation of adding at most two independent edges connecting some of the neighbors of x and deleting x. A set of five vertices C = {a, b, x, y, z} in G is said to be a crown if N G (a) = {b, x, y, z} and N G (b) = {a, x, y, z}. Given a crown C in G, reduction of C is the operation of deleting {a, b} and adding all the missing edges among {x, y, z}. In this paper, we prove that for every vertex x in a graph $$G\in{\mathcal{P}}_{3,4}$$ , there exists either (1) an edge e such that its contraction yields a graph in $${\mathcal{P}}_{3,4}$$ and at least one of its endvertices is of distance at most two from x, (2) a vertex v of degree four such that v is of distance at most two from x and some splitting at v yields a graph in $${\mathcal{P}}_{3,4}$$ , or (3) a crown C containing x whose reduction yields a graph in $${\mathcal{P}}_{3,4}$$ .

Long paths and cycles in tough graphs

For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let ?(t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let ?(t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that ?(t,n)?A·log(n) and ?(t,n)·log(?(t,n))?B·log(n). Examples are presented showing that fort≤1 there exist constantsA?, B? such that ?(t,n)≤A?·log(n) and ?(t,n)≤B?· log(n). It is conjectured that ?(t,n) ?B?·log(n) for some constantB?. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK 1.l-free graphs, wherel is fixed.