11 research outputs found
Independent Dominating Sets and a Second Hamiltonian Cycle in Regular Graphs
AbstractIn 1975, John Sheehan conjectured that every Hamiltonian 4-regular graph has a second Hamiltonian cycle. Combined with earlier results this would imply that every Hamiltonianr-regular graph (r⩾3) has a second Hamiltonian cycle. We shall verify this forr⩾300
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
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
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Extremal and Structural Problems of Graphs
In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values.
We begin in Chapter~ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance . The main result of Chapter~ comes close to proving this conjecture. Moreover, in Chapter~, we completely answer the previous question for the class of planar graphs.
Next, in Chapter~, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph must have to guarantee that, for any two-colouring of , we can partition into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours.
The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~. Given a graph and a set of colours (for some integer ), we define to be the minimum number of -coloured edges in a graph on vertices which does not contain a rainbow copy of but the addition of any non-edge in any colour from creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of , as a function of , for every connected graph of minimum degree greater than and for every integer .
In Chapter~, we consider the following question: under what conditions does a Hamiltonian graph on vertices possess a second cycle of length at least ?
We prove that the `weak' assumption of a minimum degree greater or equal to guarantees the existence of such a long cycle.
We solve two problems related to majority colouring in Chapter~. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number , the smallest positive integer such that every digraph can be coloured with colours, where each vertex has the same colour as at most a proportion of of its out-neighbours. Our main theorem states that .
We study the following problem, raised by Caro and Yuster, in Chapter~. Does every graph contain a `large' induced subgraph which has vertices of degree exactly ? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every , there exists such that any vertex graph with maximum degree contains an induced subgraph with at least vertices such that contains at least vertices of the same degree . This result is sharp up to the order of .
%Subsequently, we investigate a concept called . A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on vertices must possess a vertex of degree linear in . Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus.
Finally, in Chapter~, we move on to examine -linked tournaments. A tournament is said to be -linked if for any two disjoint sets of vertices and there are directed vertex disjoint paths such that joins to for . We prove that any strongly-connected tournament with sufficiently large minimum out-degree is -linked. This result comes close to proving a conjecture of Pokrovskiy