Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. 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 $n^{2/3}\ll m-n/2\ll n$.
Recently it was shown that a certain class of phylogenetic networks, called level-$2$ networks, cannot be reconstructed from their associated distance matrices. In this paper, we show that they can be reconstructed from their induced shortest and longest distance matrices. That is, if two level-$2$ networks induce the same shortest and longest distance matrices, then they must be isomorphic. We further show that level-$2$ networks are reconstructible from their shortest distance matrices if and only if they do not contain a subgraph from a family of graphs. A generator of a network is the grap…
Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs.
Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.
Let $G$ be a directed planar graph of complexity $n$, each arc having a nonnegative length. Let $s$ and $t$ be two distinct faces of $G$; let $s_1,...,s_k$ be vertices incident with $s$; let $t_1,...,t_k$ be vertices incident with $t$. We give an algorithm to compute $k$ pairwise vertex-disjoint paths connecting the pairs $(s_i,t_i)$ in $G$, with minimal total length, in $O(kn\log n)$ time.
Let $L_{c,n}$ denote the size of the longest cycle in $G(n,{c}/{n})$, $c>1$ constant. We show that there exists a continuous function $f(c)$ such that $ L_{c,n}/n \to f(c)$ a.s. for $c\geq 20$, thus extending a result of the author and Frieze to smaller values of $c$. Thereafter, for $c\geq 20$, we determine the limit of the probability that $G(n,c/n)$ contains cycles of every length between the length of its shortest and its longest cycles as $n\to \infty$.
Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput, and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all hamiltonian 4-regular graphs of suffi…
Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough maximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995), 301--305]. In addition, we present two generalizations of a similar result of Tkáč who considered $1$-tough maximal planar graphs [Discrete Math. 154 (1996), 321--328]; we remark that one of these generalizations gives a tight upper bound. We fix a problematic argument used in the first paper.
Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hatα\log n$ edges where $\hatα$ is the unique solution of the equation $α\log(\frac{d-2}{d-1}α) - α= \frac{d-3}{d-2}$, for $α> \frac{d-1}{d-2}$.
In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen ("On the number of longest and almost longest cycles in cubic graphs", Ars Combin., 104, 307--320, 2012) about extremal graphs in these families. In particular, we find families of cyclically $5$-edge connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs $P(2n,2)$. The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes --- more precisely, the…
We show that every $k$-tree of toughness greater than $\frac{k}{3}$ is Hamilton-connected for $k \geq 3$. (In particular, chordal planar graphs of toughness greater than $1$ are Hamilton-connected.) This improves the result of Broersma et al. (2007) and generalizes the result of Böhme et al. (1999). On the other hand, we present graphs whose longest paths are short. Namely, we construct $1$-tough chordal planar graphs and $1$-tough planar $3$-trees, and we show that the shortness exponent of the class is $0$, at most $\log_{30}{22}$, respectively. Both improve the bound of Böhme et al. Further…
We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n-3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a path of length (2n-5)^{log 2/log 3}, and …
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.
An even (respectively, odd) hole in a graph is an induced cycle with even (respectively, odd) length that is at least four. Bienstock [DM 1991 and 1992] proved that detecting an even (respectively, odd) hole containing a given vertex is NP-complete. Conforti, Chornuéjols, Kappor, and Vušković [FOCS 1997] gave the first known polynomial-time algorithm to determine whether a graph contains even holes. Chudnovsky, Kawarabayashi, and Seymour [JGT 2005] estimated that Conforti et al.'s algorithm runs in $O(n^{40})$ time on an $n$-vertex graph and reduced the required time to $O(n^{31})$. Subsequent…
Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but there are 3-connected $n$-vertex planar graphs whose longest cycles have length $Θ(n^{\log_32})$. On the other hand, Jackson and Wormald in 1992 proved that an essentially 4-connected $n$-vertex planar graph contains a cycle of length at least $(2n+4)/5$, which was recently improved to $5(n+2)/8$ by Fabrici {\it et al}. In this paper, we improve this bound to $\lceil (2n+6)/3\rceil$ for $n\ge 6$, which is best possible, by proving a quantitative version of a result of Thomassen on Tutte paths.
Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that lpt(G) \leq ceiling(n/4-n^{2/3}/90), if G is connected, lct(G) \leq ceiling(n/3-n^{2/3}/36), if G is 2-connected, and \lpt(G) \leq 3, if G is a connected circular arc graph. Our bound on lct(G) improves an earlier result of Thomassen and our bound for circular arc graphs relates to an earlier statement of Balister \emph{et al.} the argument of which contains a gap. Furthermore, we p…
We provide bounds for the product of the lengths of distinguished shortest paths in a finite network induced by a triangulation of a topological planar quadrilateral.
Given a set of paths $P$ we define the \emph{Path Covering with Forest Number} of $P$} (PCFN($P$)) as the minimum size of a set $F$ of forests satisfying that every path in $P$ is contained in at least one forest in $F$. We show that PCFN($P$) is treatable when $P$ is a set of non-crossing shortest paths in a plane graph or subclasses. We prove that if $P$ is a set of non-crossing shortest paths of a planar graph $G$ whose extremal vertices lie on the same face of $G$, then PCFN($P$)\leq 4$, and this bound is tight.
Without imposing restrictions on a weighted graph's arc lengths, symmetry structures cannot be expected. But, they exist. To find them, the graphs are decomposed into a component that dictates all closed path properties (e.g., shortest and longest paths), and a superfluous component that can be removed. The simpler remaining graph exposes inherent symmetry structures that form the basis for all closed path properties. For certain asymmetric problems, the symmetry is that of three-cycles; for the general undirected setting it is a type of four-cycles; for general directed problems with asymmetr…
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a \emph{local girth list assignment}: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thoma…
The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph. In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist $n$-vertex planar graphs whose planar edge-length ratio is in $Ω(n)$; this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and…
A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if, for every 3-separator $S$, at least one of the two components of $G-S$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm circ}\nolimits(G)$ of a longest cycle of any essentially 4-connected planar graph $G$ on $n$ vertices is at least $\frac{2n+4}{5}$ and Fabrici, Harant and Jendrol' improved this result to $\mathop{\rm circ}\nolimits(G)\geq \frac{1}{2}(n+4)$. In the present paper, we prove that an essentially 4-connected planar graph on $n$ vertices contains a cycle of length at least $\…
In this paper, we study two problems related to planar matchings in random bipartite graphs. First, we colour each edge of the complete bipartite graph $K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$ grows linearly with ${n}$ then the maximum rainbow matching is a non-trivial fraction of ${r}$ with high probability, i.e. with probability converging to one as ${n \rightarrow \infty}$ Next we consider planar matchings in a dependent setting where each vertex is forced to choose exactly one neighbour from amongst all possible choices. We obtain estimates for the lar…
We address the following question: When a randomly chosen regular bipartite multi--graph is drawn in the plane in the ``standard way'', what is the distribution of its maximum size planar matching (set of non--crossing disjoint edges) and maximum size planar subgraph (set of non--crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of $r$-regular bipartite multi--graphs with maximum planar matching (maximum planar subgraph)of at most $…
We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound Chakrabarti, Jaffe, Lee, and Vincent for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair. This also improves the largest known gap for planar graphs from 3/2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the {\em coarse differentiation} method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path me…
We address the following question: When a randomly chosen regular bipartite multi--graph is drawn in the plane in the ``standard way'', what is the distribution of its maximum size planar matching (set of non--crossing disjoint edges) and maximum size planar subgraph (set of non--crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of r-regular bipartite multi--graphs with maximum planar matching (maximum planar subgraph) of at most d …
Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\mathcal T}_t$ be the tree produced after $t$ steps. We show that there exists a fixed $δ<1$ depending on $d$ and $r$ such that almost surely for all large $t$, every $r$-ary subtree of ${\mathcal T}_t$ has less than $t^δ$ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangu…
A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.
Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of plan…
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest pat…
