82,518 research outputs found
Generalized Gr\"otzsch Graphs
The aim of this paper is to present a generalization of Gr\"otzsch graph.
Inspired by structure of the Gr\"otzsch's graph, we present constructions of
two families of graphs, and for odd and even values of
respectively and on vertices. We show that each member of this
family is non-planar, triangle-free, and Hamiltonian. Further, when is odd
the graph is maximal triangle-free, and when is even, the addition of
exactly edges makes the graph maximal triangle-free. We
show that is 4-chromatic and is 3-chromatic for all . Further,
we note some other properties of these graphs and compare with Mycielski's
construction.Comment: This is a first draft report about ongoing work on the Gr\"otzsch
Graph
Fast Recognition of Partial Star Products and Quasi Cartesian Products
This paper is concerned with the fast computation of a relation on the
edge set of connected graphs that plays a decisive role in the recognition of
approximate Cartesian products, the weak reconstruction of Cartesian products,
and the recognition of Cartesian graph bundles with a triangle free basis.
A special case of is the relation , whose convex closure
yields the product relation that induces the prime factor
decomposition of connected graphs with respect to the Cartesian product. For
the construction of so-called Partial Star Products are of particular
interest. Several special data structures are used that allow to compute
Partial Star Products in constant time. These computations are tuned to the
recognition of approximate graph products, but also lead to a linear time
algorithm for the computation of for graphs with maximum bounded
degree.
Furthermore, we define \emph{quasi Cartesian products} as graphs with
non-trivial . We provide several examples, and show that quasi
Cartesian products can be recognized in linear time for graphs with bounded
maximum degree. Finally, we note that quasi products can be recognized in
sublinear time with a parallelized algorithm
Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs
Given a graph and a graph property we say that is minimal with
respect to if no proper induced subgraph of has the property . An
HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph
, a graph is -free if has no induced subgraph isomorphic to .
The main motivation for this paper originates from a theorem of Duffus, Gould,
and Jacobson (1981), which characterizes all the minimal connected graphs with
no Hamiltonian path. In 1998, Brousek characterized all the claw-free
HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all
(not only the minimal) 2-connected non-Hamiltonian -free graphs. Recently, Cheriyan, Hajebi, and two of us (2022),
characterized all triangle-free HC-obstructions and all the HC-obstructions
which are split graphs. A wheel is a graph obtained from a cycle by adding a
new vertex with at least three neighbors in the cycle. In this paper we
characterize all the HC-obstructions which are wheel-free graphs
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