12,353 research outputs found
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
Vertex Cover Gets Faster and Harder on Low Degree Graphs
The problem of finding an optimal vertex cover in a graph is a classic
NP-complete problem, and is a special case of the hitting set question. On the
other hand, the hitting set problem, when asked in the context of induced
geometric objects, often turns out to be exactly the vertex cover problem on
restricted classes of graphs. In this work we explore a particular instance of
such a phenomenon. We consider the problem of hitting all axis-parallel slabs
induced by a point set P, and show that it is equivalent to the problem of
finding a vertex cover on a graph whose edge set is the union of two
Hamiltonian Paths. We show the latter problem to be NP-complete, and we also
give an algorithm to find a vertex cover of size at most k, on graphs of
maximum degree four, whose running time is 1.2637^k n^O(1)
Assessing the Computational Complexity of Multi-Layer Subgraph Detection
Multi-layer graphs consist of several graphs (layers) over the same vertex
set. They are motivated by real-world problems where entities (vertices) are
associated via multiple types of relationships (edges in different layers). We
chart the border of computational (in)tractability for the class of subgraph
detection problems on multi-layer graphs, including fundamental problems such
as maximum matching, finding certain clique relaxations (motivated by community
detection), or path problems. Mostly encountering hardness results, sometimes
even for two or three layers, we can also spot some islands of tractability
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