725 research outputs found
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
Characterising and recognising game-perfect graphs
Consider a vertex colouring game played on a simple graph with
permissible colours. Two players, a maker and a breaker, take turns to colour
an uncoloured vertex such that adjacent vertices receive different colours. The
game ends once the graph is fully coloured, in which case the maker wins, or
the graph can no longer be fully coloured, in which case the breaker wins. In
the game , the breaker makes the first move. Our main focus is on the
class of -perfect graphs: graphs such that for every induced subgraph ,
the game played on admits a winning strategy for the maker with only
colours, where denotes the clique number of .
Complementing analogous results for other variations of the game, we
characterise -perfect graphs in two ways, by forbidden induced subgraphs
and by explicit structural descriptions. We also present a clique module
decomposition, which may be of independent interest, that allows us to
efficiently recognise -perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the
International Colloquium on Graph Theory (ICGT) 201
Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs
Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected
graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax
the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type
degree conditions on these induced subgraphs. Let be a graph on
vertices and be an induced subgraph of . is called \emph{o}-heavy if
there are two nonadjacent vertices in with degree sum at least , and is
called -heavy if for every two vertices ,
implies that . We say that is -\emph{o}-heavy
(-\emph{f}-heavy) if every induced subgraph of isomorphic to is
\emph{o}-heavy (\emph{f}-heavy). In this paper we characterize all connected
graphs and other than such that every 2-connected
-\emph{f}-heavy and -\emph{f}-heavy (-\emph{o}-heavy and
-\emph{f}-heavy, -\emph{f}-heavy and -free) graph is Hamiltonian. Our
results extend several previous theorems on forbidden subgraph conditions and
heavy subgraph conditions for Hamiltonicity of 2-connected graphs.Comment: 21 pages, 2 figure
Ore-degree threshold for the square of a Hamiltonian cycle
A classic theorem of Dirac from 1952 states that every graph with minimum
degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured
that every graph with minimum degree at least 2n/3 contains the square of a
Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's
theorem by proving that every graph with for every contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type
version of P\'osa's conjecture for graphs on vertices using the
regularity--blow-up method; consequently the is very large (involving a
tower function). Here we present another proof that avoids the use of the
regularity lemma. Aside from the fact that our proof holds for much smaller
, we believe that our method of proof will be of independent interest.Comment: 24 pages, 1 figure. In addition to some fixed typos, this updated
version contains a simplified "connecting lemma" in Section 3.
Proper connection number of graphs
The concept of \emph{proper connection number} of graphs is an extension of proper colouring and is motivated by rainbow connection number of graphs. Let be an edge-coloured graph. Andrews et al.\cite{Andrews2016} and, independently, Borozan et al.\cite{Borozan2012} introduced the concept of proper connection number as follows: A coloured path in an edge-coloured graph is called a \emph{properly coloured path} or more simple \emph{proper path} if two any consecutive edges receive different colours. An edge-coloured graph is called a \emph{properly connected graph} if every pair of vertices is connected by a proper path. The \emph{proper connection number}, denoted by , of a connected graph is the smallest number of colours that are needed in order to make properly connected. Let be an integer. If every two vertices of an edge-coloured graph are connected by at least proper paths, then is said to be a \emph{properly -connected graph}. The \emph{proper -connection number} , introduced by Borozan et al. \cite{Borozan2012}, is the smallest number of colours that are needed in order to make a properly -connected graph.
The aims of this dissertation are to study the proper connection number and the proper 2-connection number of several classes of connected graphs. All the main results are contained in Chapter 4, Chapter 5 and Chapter 6.
Since every 2-connected graph has proper connection number at most 3 by Borozan et al. \cite{Borozan2012} and the proper connection number of a connected graph equals 1 if and only if is a complete graph by the authors in \cite{Andrews2016, Borozan2012}, our motivation is to characterize 2-connected graphs which have proper connection number 2. First of all, we disprove Conjecture 3 in \cite{Borozan2012} by constructing classes of 2-connected graphs with minimum degree that have proper connection number 3. Furthermore, we study sufficient conditions in terms of the ratio between the minimum degree and the order of a 2-connected graph implying that has proper connection number 2. These results are presented in Chapter 4 of the dissertation.
In Chapter 5, we study proper connection number at most 2 of connected graphs in the terms of connectivity and forbidden induced subgraphs , where are three integers and (where is the graph consisting of three paths with and edges having an end-vertex in common).
Recently, there are not so many results on the proper -connection number , where is an integer. Hence, in Chapter 6, we consider the proper 2-connection number of several classes of connected graphs. We prove a new upper bound for and determine several classes of connected graphs satisfying . Among these are all graphs satisfying the Chv\'{a}tal and Erd\'{o}s condition ( with two exceptions). We also study the relationship between proper 2-connection number and proper connection number of the Cartesian product of two nontrivial connected graphs.
In the last chapter of the dissertation, we propose some open problems of the proper connection number and the proper 2-connection number
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