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A connected graph is 4-connected if it contains at least five vertices and removing any three of them does not disconnect it. A frequent preprocessing step in graph drawing is to decompose a plane graph into its 4-connected components and…
In this study, we take a systematic look at the unrealised part of public transport networks (PTNs) with functional connections. We consider their complement graphs and study their structure. The complement graph $\bar G$ of an unweighted…
Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and…
According to the so-called strong version of structural balance theory, actors in signed social networks avoid establishing triads with an odd number of negative links. Generalising, the weak version of balance theory allows for nodes to be…
The {\em disjointness graph} of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph $G$ of any system of segments in the plane is {\em…
We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.
In this research, we determine the structure of (claw, bull)-free graphs. We show that every connected (claw, bull)-free graph is either an expansion of a path, an expansion of a cycle, or the complement of a triangle-free graph; where an…
We give a new proof of the theorem of Boesch-Tindell and Farzad-Mahdian-Mahmoodian-Saberi-Sadri that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. Our proof…
Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of…
The Bar\'at-Thomassen conjecture asserts that there is a function $f$ such that for every fixed tree $T$ with $t$ edges, every graph which is $f(t)$-edge-connected with its number of edges divisible by $t$ has a partition of its edges into…
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…
In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a…
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.
If a biconnected graph stays connected after the removal of an arbitrary vertex and an arbitrary edge, then it is called 2.5-connected. We prove that every biconnected graph has a canonical decomposition into 2.5-connected components. These…
The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of…
We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to…
Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a…
We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $\delta_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $\delta_2(H) \geq 4n/5 + o(n)$…
In his article [J. Comb. Theory Ser. B 16 (1974), 168-174], Tutte called two graphs $T$-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs $G$ and $G'$ are $T$-equivalent if $G'$ is obtained from…
We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct…