Goldberg Snarks Research Articles

Computational results and new bounds for the circular flow number of snarks

It is well known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to make an efficient and practical implementation of an algorithm for the computation of the circular flow number of a bridgeless cubic graph. Using this procedure, we determine the circular flow number of all snarks on up to 36 vertices as well as the circular flow number of various famous snarks. After that, as combination of the use of this algorithm with new theoretical results, we present an infinite family of snarks of order 8k+2 whose circular flow numbers meet a general lower bound presented by Lukot’ka and Škoviera in 2008. In particular this answers a question proposed in their paper. Moreover, we improve the best known upper bound for the circular flow number of Goldberg snarks and we conjecture that this new upper bound is optimal.

Sigma Coloring on Powers of Paths and Some Families of Snarks

Consider a vertex coloring of a graph where each color is represented by a natural number. The color sum of a vertex is the sum of the colors of its adjacent vertices. The Sigma Coloring Problem concerns determining the sigma chromatic number of a graph G, σ(G), which is the least number of colors for a coloring of G such that the color sum of any two adjacent vertices are different. In this article, we proved that σ(Pnk)≤3 when 2≤k≤n3−1, and we determined the sigma chromatic number for Pnk in the remaining cases. This article also presents the sigma chromatic number for Blanuša 1st and 2nd families of snarks, Flower snarks, Goldberg and Twisted Goldberg snarks.

Feedback Numbers of Goldberg Snark, Twisted Goldberg Snarks and Related Graphs

A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. The minimum cardinality of a feedback vertex set is called the feedback number. In this paper, we determine the exact values of the feedback numbers of the Goldberg snarks Gn and its related graphs Gn*, Twisted Goldberg Snarks TGn and its related graphs TGn*. Let f(n) denote the feedback numbers of these graphs, we prove that f(n)=2n+1, for n≥3.

The hunting of a snark with total chromatic number 5

A snark is a cyclically-4-edge-connected cubic graph with chromatic index 4. In 1880, Tait proved that the Four-Color Conjecture is equivalent to the statement that every planar bridgeless cubic graph has chromatic index 3. The search for counter-examples to the Four-Color Conjecture motivated the definition of the snarks.A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent or incident elements have different colors. The total chromatic number ofG, denoted by χT(G), is the least k for which G has a k-total-coloring. Rosenfeld has shown that the total chromatic number of a cubic graph is either 4 or 5. However, the problem of determining the total chromatic number of a graph is NP-hard even for cubic bipartite graphs.In 2003, Cavicchioli et al. reported that their extensive computer study of snarks shows that all square-free snarks with less than 30 vertices have total chromatic number 4, and asked for the smallest order of a square-free snark with total chromatic number 5.In this paper, we prove that the total chromatic number of both Blanuša’s families and an infinite square-free snark family (including the Loupekhine and Goldberg snarks) is 4. Relaxing any of the conditions of cyclic-edge-connectivity and chromatic index, we exhibit cubic graphs with total chromatic number 5.

Petersen-colorings and some families of snarks

In this paper we study Petersen-colorings and strong Petersen-colorings on some well known families of snarks, e.g. Blanuša snarks, Goldberg snarks and flower snarks. In particular, it is shown that flower snarks have a Petersen-coloring but they do not have a strong Petersen-coloring. Furthermore it is proved that possible minimum counterexamples to Jaeger’s Petersen-coloring conjecture do not contain a specific subdivision of K 3, 3 .

Circular flow number of generalized Blanuša snarks

A circular flow on a graph is an assignment of directions and flow values from R to the edges so that for each vertex the sum of the flow values on exiting edges equals the sum of the flow values on entering edges. A circular nowhere-zeror-flow is a circular flow ϕ with flow values satisfying 1≤|ϕ(e)|≤r−1 for each edge e. The circular flow number of a graph G is the infimum of all reals r such that G has a circular nowhere-zero r-flow. We prove that the circular flow number of all generalized Blanuša snarks except for the Petersen graph is 4.5. We also bound the circular flow number of Goldberg snarks, both from above and from below.

The total-chromatic number of some families of snarks

The total-chromatic number χ T ( G ) is the least number of colours needed to colour the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same colour. It is known that the problem of determining the total-chromatic number is NP-hard, and it remains NP-hard even for cubic bipartite graphs. Snarks are simple connected bridgeless cubic graphs that are not 3-edge-colourable. In this paper, we show that the total-chromatic number is 4 for three infinite families of snarks, namely, the Flower Snarks, the Goldberg Snarks, and the Twisted Goldberg Snarks. This result reinforces the conjecture that all snarks have total-chromatic number 4. Moreover, we give recursive procedures to construct a total-colouring that uses 4 colours in each case.

Computing the Automorphic Chromatic Index of Certain Snarks

The automorphic G-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by the full automorphism group G of Γ. We determine the automorphic G-chromatic index of each member of four infinite classes of snarks: type I Blanusa snarks, type II Blanusa snarks, Flower snarks and Goldberg snarks.

A note on Berge–Fulkerson coloring

The Berge–Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graph G admits a Berge–Fulkerson coloring if and only if the graph G contains a pair of edge-disjoint matchings M 1 and M 2 such that (i) M 1 ∪ M 2 induces a 2- regular subgraph of G and (ii) the suppressed graph G ∖ M i ¯ , the graph obtained from G ∖ M i by suppressing all degree-2- vertices, is 3- edge-colorable for each i = 1 , 2 . This lemma is further applied in the verification of Berge–Fulkerson Conjecture for some families of non-3-edge-colorable cubic graphs (such as, Goldberg snarks, flower snarks).

Infinite Classes of Dihedral Snarks

Flower snarks and Goldberg snarks are two infinite families of cyclically 5–edge–connected cubic graphs with girth at least five and chromatic index four. For any odd integer k, k > 3, there is a Flower snark, say Jk, of order 4k and a Goldberg snark, say Bk, of order 8k. We determine the automorphism groups of Jk and Bk for every k and prove that they are isomorphic to the dihedral group D4k of order 4k.

The circular chromatic index of Goldberg snarks

We determine the exact values of the circular chromatic index of the Goldberg snarks, and of a related family, the twisted Goldberg snarks.