Related papers: The Complexity of Color-constrained Paths in Semic…
Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that…
The complexity of {\sc Colouring} is fully understood for $H$-free graphs, but there are still major complexity gaps if two induced subgraphs $H_1$ and $H_2$ are forbidden. Let $H_1$ be the $s$-vertex cycle $C_s$ and $H_2$ be the $t$-vertex…
We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The…
A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates…
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every…
We study the following generalization of the Hamiltonian cycle problem: Given integers $a,b$ and graph $G$, does there exist a closed walk in $G$ that visits every vertex at least $a$ times and at most $b$ times? Equivalently, does there…
In recent work by Johnson et al. (2022), a framework was described for the study of graph problems over classes specified by omitting each of a finite set of graphs as subgraphs. If a problem falls into the framework then its computational…
In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete in planar bipartite graphs with…
In this paper we present the first deterministic polynomial time algorithm for determining the existence of a Hamiltonian cycle and finding a Hamiltonian cycle in general graphs. Our algorithm can also solve the Hamiltonian path problem in…
Given a $c$-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an…
Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…
Motivated by a conjecture of Liang [Y.-C. Liang. {\em Anti-magic labeling of graphs}. PhD thesis, National Sun Yat-sen University, 2013.], we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free…
The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was…
Given an edge-colored graph, the Maximum Rainbow Matching problem asks for a maximum-cardinality matching of the graph that contains at most one edge from each color. We provide the following complexity dichotomy for this problem based on…
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering…
We introduce a new bilevel version of the classic shortest path problem and completely characterize its computational complexity with respect to several problem variants. In our problem, the leader and the follower each control a subset of…
For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal…
Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is…
We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden…
We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…