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

The total zero-divisor graph of commutative rings

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graphs of the ring ${\mathbb Z}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb Z}_m$ and ${\mathbb Z}_n$ are isomorphic if and only if $m=n$

Uniquely hamiltonian graphs for many sets of degrees

We give constructive proofs for the existence of uniquely hamiltonian graphs for various sets of degrees. We give constructions for all sets with minimum 2 (a trivial case), all sets with minimum 3 that contain an even number (for sets without an even number it is known that no uniquely hamiltonian graphs exist), and all sets with at least two elements and minimum 4 where all other elements are at least 10. For minimum degree 3 and 4, the constructions also give 3-connected graphs

Stable dominating circuits in snarks

AbstractSnarks are cyclically 4-edge-connected cubic graphs with girth at least 5 and with no 3-edge-coloring. We construct snarks with a (dominating) circuit C so that no other circuit C′ satisfies V(C)⊆V(C′). These graphs are of interest because two known conjectures about graphs can be reduced on them. The first one is Sabidusi's Compatibility Conjecture which suggests that given an eulerian trail T in an eulerian graph G without 2-valent vertices, there exists a decomposition of G into circuits such that consecutive edges in T belong to different circuits. The second conjecture is the Fixed-Circuit Cycle Double-Cover Conjecture suggesting that every bridgeless graph has a cycle double cover which includes a fixed circuit

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201

Graphs with few Hamiltonian Cycles

We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number $k \ge 0$ of hamiltonian cycles, which is especially efficient for small $k$. 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 $n$ iff $n \ge 18$ 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 $3$ 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 $48$ 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 $n$, the exact number of such graphs on $n$ vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio

Perfect Matching and Circuit Cover of Graphs

The research of my dissertation is motivated by the Circuit Double Cover Conjecture due to Szekeres and independently Seymour, that every bridgeless graph G has a family of circuits which covers every edge of G twice. By Fleischner\u27s Splitting Lemma, it suffices to verify the circuit double cover conjecture for bridgeless cubic graphs.;It is well known that every edge-3-colorable cubic graph has a circuit double cover. The structures of edge-3-colorable cubic graphs have strong connections with the circuit double cover conjecture. In chapter two, we consider the structure properties of a special class of edge-3-colorable cubic graphs, which has an edge contained by a unique perfect matching. In chapter three, we prove that if a cubic graph G containing a subdivision of a special class of edge-3-colorable cubic graphs, semi-Kotzig graphs, then G has a circuit double cover.;Circuit extension is an approach posted by Seymour to attack the circuit double cover conjecture. But Fleischer and Kochol found counterexamples to this approach. In chapter four, we post a modified approach, called circuit extension sequence. If a cubic graph G has a circuit extension sequence, then G has a circuit double cover. We verify that all Fleischner\u27s examples and Kochol\u27s examples have a circuit extension sequence, and hence not counterexamples to our approach. Further, we prove that a circuit C of a bridgeless cubic G is extendable if the attachments of all odd Tutte-bridges appear on C consequently.;In the last chapter, we consider the properties of minimum counterexamples to the strong circuit double cover. Applying these properties, we show that if a cubic graph G has a long circuit with at least | V(G)| - 7 vertices, then G has a circuit double cover