6,939 research outputs found
Graphs with few Hamiltonian Cycles
We describe an algorithm for the exhaustive generation of non-isomorphic
graphs with a given number of hamiltonian cycles, which is especially
efficient for small . Our main findings, combining applications of this
algorithm and existing algorithms with new theoretical results, revolve around
graphs containing exactly one hamiltonian cycle (1H) or exactly three
hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work
of Royle, we show that there exist nearly cubic 1H graphs of order iff is even. This gives the strongest form of a theorem of Entringer and
Swart, and sheds light on a question of Fleischner originally settled by
Seamone. We prove equivalent formulations of the conjecture of Bondy and
Jackson that every planar 1H graph contains two vertices of degree 2, verify it
up to order 16, and show that its toric analogue does not hold. We treat
Thomassen's conjecture that every hamiltonian graph of minimum degree at least
contains an edge such that both its removal and its contraction yield
hamiltonian graphs. We also verify up to order 21 the conjecture of Sheehan
that there is no 4-regular 1H graph. Extending work of Schwenk, we describe all
orders for which cubic 3H triangle-free graphs exist. We verify up to order
Cantoni's conjecture that every planar cubic 3H graph contains a triangle,
and show that there exist infinitely many planar cyclically 4-edge-connected
cubic graphs with exactly four hamiltonian cycles, thereby answering a question
of Chia and Thomassen. Finally, complementing work of Sheehan on 1H graphs of
maximum size, we determine the maximum size of graphs containing exactly one
hamiltonian path and give, for every order , the exact number of such graphs
on vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio
Hamilton cycles in almost distance-hereditary graphs
Let be a graph on vertices. A graph is almost
distance-hereditary if each connected induced subgraph of has the
property for any pair of vertices .
A graph is called 1-heavy (2-heavy) if at least one (two) of the end
vertices of each induced subgraph of isomorphic to (a claw) has
(have) degree at least , and called claw-heavy if each claw of has a
pair of end vertices with degree sum at least . Thus every 2-heavy graph is
claw-heavy. In this paper we prove the following two results: (1) Every
2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian.
(2) Every 3-connected, 1-heavy and almost distance-hereditary graph is
Hamiltonian. In particular, the first result improves a previous theorem of
Feng and Guo. Both results are sharp in some sense.Comment: 14 pages; 1 figure; a new theorem is adde
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
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
On random k-out sub-graphs of large graphs
We consider random sub-graphs of a fixed graph with large minimum
degree. We fix a positive integer and let be the random sub-graph
where each independently chooses random neighbors, making
edges in all. When the minimum degree then is -connected w.h.p. for ;
Hamiltonian for sufficiently large. When , then has
a cycle of length for . By w.h.p. we mean
that the probability of non-occurrence can be bounded by a function
(or ) where
Heavy subgraphs, stability and hamiltonicity
Let be a graph. Adopting the terminology of Broersma et al. and \v{C}ada,
respectively, we say that is 2-heavy if every induced claw () of
contains two end-vertices each one has degree at least ; and
is o-heavy if every induced claw of contains two end-vertices with degree
sum at least in . In this paper, we introduce a new concept, and
say that is \emph{-c-heavy} if for a given graph and every induced
subgraph of isomorphic to and every maximal clique of ,
every non-trivial component of contains a vertex of degree at least
in . In terms of this concept, our original motivation that a
theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and
-c-heavy graph is hamiltonian, where is the graph obtained from a
triangle by adding three disjoint pendant edges. In this paper, we will
characterize all connected graphs such that every 2-connected o-heavy and
-c-heavy graph is hamiltonian. Our work results in a different proof of a
stronger version of Hu's theorem. Furthermore, our main result improves or
extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones
Mathematicae Graph Theor
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