Related papers: On traceable iterated line graph and hamiltonian p…
Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic…
We prove that for every $\varepsilon > 0$ there is $c_0$ such that if $G\sim G(n,c/n)$, $c\ge c_0$, then with high probability $G$ can be covered by at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ vertex disjoint paths, which is…
Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amount of research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, and to related questions. The corresponding…
Given a graph $G=(V,E)$, the longest induced path problem asks for a maximum cardinality node subset $W\subseteq V$ such that the graph induced by $W$ is a path. It is a long established problem with applications, e.g., in network analysis.…
We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus…
Recently, Letzter proved that any graph of order $n$ contains a collection $\mathcal{P}$ of $O(n\log^\star n)$ paths with the following property: for all distinct edges $e$ and $f$ there exists a path in $\mathcal{P}$ which contains $e$ but…
Following a problem posed by Lov\'asz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that…
A spanning subgraph of a graph G is called a [0,2]-factor of G, if for . is a union of some disjoint cycles, paths and isolate vertices, that span the graph G. It is easy to get a [0,2]-factor of G and there would be many of [0,2]-factors…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code…
A path $P$ in a graph $G$ is said to be a degree monotone path if the sequence of degrees of the vertices of $P$ in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by…
We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n=1. This theorem counts the number of bound states in terms of…
Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181]…
A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at…
We investigate locally $n \times n$ grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on $n$ vertices. We consider the subclass of these graphs for which each pair of…
A path of a graph $G$ is called a Hamilton path if it passes through all the vertices of $G$. A graph is Hamilton-connected if any two vertices are connected by a Hamilton path. Note that any bipartite graph is not Hamilton-connected. We…
A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such…
A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2.…
We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree…
Let $G$ be a graph on $n\geq 3$ vertices. A graph $G$ is almost distance-hereditary if each connected induced subgraph $H$ of $G$ has the property $d_{H}(x,y)\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\in V(H)$. A graph $G$ is called…
A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an…