MAX CUT in cubic graphs

We present an improved semidefinite programming based approximation algorithm for the MAX CUT problem in graphs of maximum degree at most 3. The approximation ratio of the new algorithm is at least 0.9326. This improves, and also somewhat simplifies, a result of Feige, Karpinski and Langberg. We also observe that results of Hopkins and Staton and of Bondy and Locke yield a simple combinatorial 4/5-approximation algorithm for the problem. Finally, we present a combinatorial 22/27-approximation algorithm for the MAX CUT problem for regular cubic graphs.     

Numbers of cubic graphs

The numbers of unlabeled cubic graphs on p = 2n points have been found by two different counting methods, the best of which has given values for p ≦ 40.     

Gromov hyperbolic cubic graphs

If X is a geodesic metric space and x ₁; x ₂; x ₃ ∈ X, a geodesic triangle T = {x ₁; x ₂; x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.     

Change graphs of edge-colorings of planar cubic graphs

It will be proved that the number of vertices of each component of the change-graph of two edge-colorings of an arbitrary planar cubic graph is even (here a change-graph is the subgraph containing exactly those edges having different colors in the considered two edge-colorings and moreover only those vertices which are incident with at least one of these edges).     

On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let ${\nu }_i\left(G\right)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that ${\nu }_2\left(G\right)\ge \frac{4}{5}\left|V\left(G\right)\right|$ and ${\nu }_3\left(G\right)\ge \frac{7}{6}\left|V\left(G\right)\right|$. Moreover, it turns out that ${\nu }_2\left(G\right)\le \frac{\left|V\left(G\right)\right|+2{\nu }_3\left(G\right)}{4}$.     

Counting labeled general cubic graphs

Recurrence relations are derived for the numbers of labeled 3-regular graphs with given connectivity, order, number of double edges, and number of loops. This work builds on methods previously developed by Read, Wormald, Palmer, and Robinson.     

Fast generation of cubic graphs

In this paper an efficient algorithm to generate regular graphs with small vertex valency is presented. The running times of a program based on this algorithm and designed to generate cubic graphs are below two natural benchmarks: (a) If N(n) denotes the number of pairwise non-isomorphic cubic graphs with n vertices and T(n) the time needed for generating the list of all these graphs, then T(n)/N(n) decreases gradually for the observed values of n. This suggests that T(n)/N(n) might be uniformly bounded for all n, ignoring the time to write the outputs, but we are unable to prove this and in fact are not confident about it. (b) For programs that generate lists of non-isomorphic objects, but cannot a priori make sure to avoid the generation of isomorphic copies, the time needed to check a randomly ordered list of these objects for being non-isomorphic is a natural benchmark. Since for large lists (n ≥ 22, girth 3) existing graph isomorphism programs take longer to canonically label all of the N(n) graphs than our algorithm takes to generate them, our algorithm is probably faster than any method which does one or more isomorphism test for every graph. © 1996 John Wiley & Sons, Inc.     

Rainbowness of cubic plane graphs

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that

     

Bipartite density of cubic graphs

We first obtain the exact value for bipartite density of a cubic line graph on n vertices. Then we give an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of the adjacency matrix. In addition, we characterize, except in the case n=20, those graphs for which the upper bound is obtained.     

The toughness of cubic graphs

The toughness of a graphG is the minimum of|S|/k(G – S) over all setsS of vertices such thatk(G – S) 2. In this paper upper bounds on the toughness of a cubic graph are derived in terms of the independence number and coloring parameters. These are applied to cycle permutation graphs.     

Multicolored forests in complete bipartite graphs

If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=(a₁,a₂,…,a_k), in which a_i denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph K_n,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of K_n,n into multicolored trees.     

Uniform cyclic edge connectivity in cubic graphs

A cubic graph which is cyclicallyk-edge connected and has the further property that every edge belongs to some cyclick-edge cut is called uniformly cyclicallyk-edge connected(U(k)). We classify theU(5) graphs and show that all cyclically 5-edge connected cubic graphs can be generated from a small finite set ofU(5) graphs by a sequence of defined operations.MATHDAHCOTAGE.AC.NZ     

Sparsely intersecting perfect matchings in cubic graphs

In 1971, Fulkerson made a conjecture that every bridgeless cubic graph contains a family of six perfect matchings such that each edge belongs to exactly two of them; equivalently, such that no three of the matchings have an edge in common. In 1994, Fan and Raspaud proposed a weaker conjecture which requires only three perfect matchings with no edge in common. In this paper we discuss these and other related conjectures and make a step towards Fulkerson’s conjecture by proving the following result: Every bridgeless cubic graph which has a 2-factor with at most two odd circuits contains three perfect matchings with no edge in common.     

Oddness to resistance ratios in cubic graphs

Let $G$ be a bridgeless cubic graph. Oddness (weak oddness) is defined as the minimum number of odd components in a 2-factor (an even factor) of $G$, denoted as $\omega \left(G\right)$ (Steffen, 2004) (${\omega }^{\prime }\left(G\right)$ Lukot’ka and Mazák (2016)). Oddness and weak oddness have been referred to as measurements of uncolourability (Fiol et al., 2017, Lukot’ka and Mazák, 2016, Lukot’ka et al., 2015 and, Steffen, 2004), due to the fact that $\omega \left(G\right)=0$ and ${\omega }^{\prime }\left(G\right)=0$ if and only if $G$ is 3-edge-colourable. Another so-called measurement of uncolourability is resistance, defined as the minimum number of edges that can be removed from $G$ such that the resulting graph is 3-edge-colourable, denoted as $r\left(G\right)$ (Steffen, 2004). It is easily shown that $\omega \left(G\right)\ge {\omega }^{\prime }\left(G\right)\ge r\left(G\right)$. While it has been shown that the difference between any two of these measures can be arbitrarily large, it has been conjectured that ${\omega }^{\prime }\left(G\right)\le 2r\left(G\right)$, and that if $G$ is a snark then $\omega \left(G\right)\le 2r\left(G\right)$ (Fiol et al., 2017). In this paper, we disprove the latter by showing that the ratio of oddness to weak oddness can be arbitrarily large. We also offer some insights into the former conjecture by defining what we call resistance reducibility, and hypothesizing that almost all cubic graphs are such resistance reducible.     

On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G)=2 when G is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332-334] conjectured that if |V(G)|≡0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.