Using the graphs of prisms and Tutte Fragments, we construct an infinite family of hamiltonian and non-hamiltonian graphs in which Tutte's counterexample to Tait's conjecture appears in a certain sense as a minimal element. We observe that generalizations of the minimum-cardinality counterexamples of Holton and McKay to Tait's conjecture are as well contained in this family.

On the basis of recent results on hamiltonicity, [5], and hamiltonian connectedness, [9], in the square of a 2-block, we determine the most general block-cutvertex structure a graph G may have in order to guarantee that G2 is hamiltonian, hamiltonian connected, respectively. Such an approach was already developed in [10] for hamiltonian total graphs.

A perfect pseudo-matching M in a cubic graph G is a spanning subgraph of G such that every component of M is isomorphic to K_2 or to K_1,3. In view of snarks G with dominating cycle C, this is a natural generalization of perfect matchings since G-E(C) is a perfect pseudo-matching. Of special interest are such M where the graph G/M is planar because such G have a cycle double cover. We show that various well known classes of snarks contain planarizing perfect pseudo-matchings, and that there are at least as many snarks with planarizing perfect pseudo-matchings as there are cyclically 5-edge-connected snarks.

In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite graphs is an NP-complete problem.

It is shown that for any choice of four different vertices x_1,...,x_4 in a 2-block G of order p>3, there is a hamiltonian cycle in G^2 containing four different edges x_iy_i of E(G) for certain vertices y_i, i=1,2,3,4. This result is best possible.

We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor Q. Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G/Q. In particular, we study the case where G is the leapfrog extension (called vertex envelope in (Discrete Math., 309(14):4793-4809, 2009)) of a plane cubic graph G_0. As a consequence we prove hamiltonicity in the leapfrog extension of planar cubic cyclically 4-edge-connected bipartite graphs. This and other results of this paper establish partial solutions of Barnette's Conjecture according to which every 3-connected cubic planar bipartite graph is hamiltonian. These results go considerably beyond Goodey's result on this topic (Israel J. Math., 22:52-56, 1975).

•New algorithms for finding uniquely hamiltonian cycles.•Transforming stable fixed edge cycles to uniquely hamiltonian cycles.•Strong properties of a minimum counter example to Bondy and Jackson’s conjecture.•Verifies Bondy and Jackson’s conjecture for graphs with up to 25 vertices.

The cycle double cover conjecture is a famous longstanding unsolved conjecture in graph theory. It is related and can be reduced to the compatible circuit decomposition problem. Recently Fleischner et al. (2018) provided a sufficient condition for a compatible circuit decomposition, which is called SUD-$$K_5$$-minor freeness. In a previous work we developed an abstract mathematical model for finding SUD-$$K_5$$-minors and based on the model a mixed integer linear program (MIP). In this work we propose a respective boolean satisfiability (SAT) model and compare it with the MIP model in computational tests. Non-trivial symmetry breaking constraints are proposed, which improve the solving times of both models considerably. Compared to the MIP model the SAT approach performs significantly better. We use the faster algorithm to further test graphs of graph theoretic interest and were able to get new insights. Among other results we found snarks with 30 and 32 vertices that do not contain a perfect pseudo-matching, that is a spanning subgraph consisting of $$K_2$$ and $$K_{1,3}$$ components, whose contraction leads to a SUD-$$K_5$$-minor free graph.

The well known cycle double cover conjecture in graph theory is strongly related to the compatible circuit decomposition problem. A recent result by Fleischner et al. (2018) gives a sufficient condition for the existence of a compatible circuit decomposition in a transitioned 2-connected Eulerian graph, which is based on an extension of the definition of K-5-minors to transitioned graphs. Graphs satisfying this condition are called SUD-K-5-minor-free graphs. In this work we formulate a generalization of this property by replacing the K-5 by a 4-regular transitioned graph H, which is part of the input. Furthermore, we consider the decision problem of checking for two given graphs if the extended property holds. We prove that this problem is NP-complete and fixed parameter tractable with the size of H as parameter. We then formulate an equivalent problem, present a mathematical model for it, and prove its correctness. This mathematical model is then translated into a mixed integer linear program (MIP) for solving it in practice. Computational results show that the MIP formulation can be solved for small instances in reasonable time. In our computations we found snarks with perfect matchings whose contraction leads to SUD-K-5-minor-free graphs that contain K-5-minors. Furthermore, we verified that there exists a perfect pseudo-matching whose contraction leads to a SUD-K-5-minor-free graph for all snarks with up to 22 vertices. (C) 2020 Elsevier B.V. All rights reserved.

This is the second part of joint research in which we show that every $2$-connected graph $G$ has the ${\cal F}_4$ property. That is, given distinct $x_i\in V(G)$, $1\leq i\leq 4$, there is an $x_1x_2$-hamiltonian path in $G^2$ containing different edges $x_3y_3, x_4y_4\in E(G)$ for some $y_3,y_4\in V(G)$. However, it was shown already in \cite[Theorem 2]{cf1:refer} that 2-connected DT-graphs have the ${\cal F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs. We also show that these results are best possible.

The cycle double cover conjecture is a famous longstanding unsolved conjecture in graph theory. It is related and can be reduced to the compatible circuit decomposition problem. Recently Fleischner et al. (2018) provided a sufficient condition for a compatible circuit decomposition, which is called SUDK5-minor freeness. In a previous work we developed an abstract mathematical model for finding SUD-K5-minors and based on the model a MIP-formulation. In this work we propose a respective SAT-model and compare it with the MIP model in computational tests. Non-trivial symmetry breaking constraints are proposed, which improve the solving times of both models considerably. Compared to the MIP model the SAT approach performs significantly better. We use the faster algorithm to further test graphs of graph theoretic interest and were able to get new insights. Among other results we found snarks with 30 and 32 vertices that do not contain a perfect pseudo-matching, that is a spanning subgraph consisting of K2 and K1,3 components, whose contraction leads to a SUD-K5-minor free graph.

Let G be a 2-connected eulerian graph. For each vertex v∈V(G), let T(v) be the set of edge-disjoint edge-pairs of E(v), and, T=⋃v∈V(G)T(v). A circuit decomposition C of G is compatible with T if |E(C)∩P|≤1 for every member C∈C and every P∈T. Fleischner (1990's) wondered implicitly whether if (G,T) does not have a compatible circuit decomposition then (G,T) must have an undecomposable K5-transition-minor or its generalized transition-minor. This long-standing open problem was partially verified for various graph-minor-free families of graphs, for example, it was solved by Fleischner for planar graphs (Fleischner (1980) [7]) and solved by Fan and Zhang for K5-minor-free graphs (Fan and Zhang (2000) [6]). This transition-minor-free conjecture is now completely solved in this paper. And, as a by-product and a necessary stepping-stone, we characterize the structure of sup-undecomposable K5-minor-free graphs (G,T) in which every compatible circuit decomposition consists of a pair of Hamiltonian circuits. This result plays an important role in the proof of the main theorem and also generalizes an earlier result by Lai and Zhang (Lai and Zhang (2001) [13]).

We show that every 2-connected cubic graph G has a cycle double cover if G has a spanning subgraph F such that (i) every component of F has an even number of vertices (ii) every component of F is either a cycle or a subdivision of a Kotzig graph and (iii) the components of F are connected to each other in a certain general manner.

It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem even if restricted to planar 3-connected eulerian graphs. In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. We show, in particular, that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite graphs is an NP-complete problem.

The square of a graph G, denoted G^2, is the graph obtained from G by joining by an edge any two nonadjacent vertices which have a common neighbor. A graph G is said to have the F_k property if for any set of k distinct vertices {x_1, x_2, ..., x_k} in G, there is a hamiltonian path from x_1 to x_2 in G^2 containing k-2 distinct edges of G of the form x_iz_i, i = 3, ..., k. It was proved many years ago that every 2-connected graph has the F_3 property. In the first part of this work, we extend this result by proving that every 2-connected DT-graph has the F_4 property (Theorem 2) and will show in the second part that this generalization holds for arbitrary 2-connected graphs, and that there exist 2-connected graphs which do not have the F_k property for any natural number k >= 5. Altogether, this answers a problem raised before in the affirmative.

The well known cycle double cover conjecture in graph theory is strongly related to the compatible circuit decomposition problem. A recent result by Fleischner et al. (2018) gives a sufficient condition for the existence of a compatible circuit decomposition in a transitioned 2-connected Eulerian graph, which is based on an extension of the definition of K5-minors to transitioned graphs. Graphs satisfying this condition are called SUD-K5-minor-free graphs. In this work we formulate a generalization of this property by replacing the K5 by a 4-regular transitioned graph H, which is part of the input. Furthermore, we consider the decision problem of checking for two given graphs if the extended property holds. We prove that this problem is NPcomplete and fixed parameter tractable with the size of H as parameter. We then formulate an equivalent problem, present a mathematical model for it, and prove its correctness. This mathematical model is then translated into a mixed integer linear program (MIP) for solving it in practice. Computational results show that the MIP formulation can be solved for small instances in reasonable time. In our computations we found snarks with perfect matchings whose contraction leads to SUD-K5-minor-free graphs that contain K5-minors. Furthermore, we verified that there exists a perfect pseudo-matching whose contraction leads to a SUD-K5-minor-free graph for all snarks with up to 22 vertices.

Viewing fullerenes as plane graphs with facial cycles being pentagonal and hexagonal only, it is shown how to reduce an arbitrary fullerene to the (graph of the) dodecahedron. This can be achieved by a sequence of eight reduction steps, seven of which are local operations and the remaining reduction step acts globally. In any case, the resulting algorithm has polynomial running time.

In this paper we formulate an algorithm for finding smooth graphs with small independence numbers. To this end we formalize a family of satisfaction problems and propose a branch-and-bound-based approach for solving them. Strong bounds are obtained by exploiting graph-theoretic aspects including new results obtained in cooperation with leading graph theorists. Based on a partial solution we derive a lower bound by computing an independent set on a partial graph and finding a lower bound on the size of possible extensions. The algorithm is used to test conjectured lower bounds on the independence numbers of smooth graphs and some subclasses of smooth graphs. In particular for the whole class of smooth graphs we test the lower bound of 2n/7 for all smooth graphs with at least n ≥ 12 vertices and can proof the correctness for all 12 ≤ n ≤ 24 . Furthermore, we apply the algorithm on different subclasses, such as all triangle free smooth graphs.

Let G be an eulerian graph. For each vertex v∈V(G), let T(v) be a non-empty subset of a partition of the edges incident with v into 2-subsets and set T=∪v∈V(G)T(v), called a transition system of G. A transition system T of G is admissible if T∩F|≤12|F| for every T∈T and every edge cut F of G. A cycle decomposition C of G is called a compatible cycle decomposition (CCD for short) of (G, T) if |E(C)∩T|≤1 for every cycle C∈C and every T∈T. H. Fleischner proved that if G is planar, then for every admissible transition system T of G, (G, T) has a CCD. G. Fan and C.-Q. Zhang (2000, J. Combin. Theory Ser. B 78, 1-23) showed that this result is also true for K5-minor-free graphs. We generalize this result to all eulerian graphs that do not contain a special type of K5-minor which is called a bad K5-minor. To this purpose, we characterise the 4−regular “hereditary” bad K5-minor-free graphs (G, T) in which every CCD of (G, T) is a pair of hamiltonian cycles.

In graph theory, a prominent conjecture of Bondy and Jackson states that every uniquely hamiltonian planar graph must have a vertex of degree two. In this work we try to find uniquely hamiltonian graphs with minimum degree three and a small crossing number by minimizing the number of crossings in an embedding and the number of degree-two vertices. We formalize an optimization problem for this purpose and propose a general variable neighborhood search (GVNS) for solving it heuristically. The several different types of used neighborhoods also include an exponentially large neighborhood that is effectively searched by means of branch and bound. To check feasibility of neighbors we need to solve hamiltonian cycle problems, which is done in a delayed manner to minimize the computation effort. We compare three different configurations of the GVNS. Although our implementation could not find a uniquely hamiltonian planar graph with minimum degree three disproving Bondy and Jackson’s conjecture, we were able to find uniquely hamiltonian graphs of minimum degree three with crossing number four for all number of vertices from 10 to 100.