Decompositions and reductions of snarks

According to M. Gardner [“Mathematical Games: Snarks, Boojums, and Other Conjectures Related to the Four-Color-Map Theorem,” Scientific American, vol. 234 (1976), pp. 126–130], a snark is a nontrivial cubic graph whose edges cannot be properly colored by three colors. The problem of what “nontrivial” means is implicitly or explicitly present in most papers on snarks, and is the main motivation of the present paper. Our approach to the discussion is based on the following observation. If G is a snark with a k-edge-cut producing components G₁ and G₂, then either one of G₁ and G₂ is not 3-edge-colorable, or by adding a “small” number of vertices to either component one can obtain snarks G₁ and G₂ whose order does not exceed that of G. The two situations lead to a definition of a k-reduction and k-decomposition of G. Snarks that for m < k do not admit m-reductions, m-decompositions, or both are k-irreducible, k-indecomposable, and k-simple, respectively. The irreducibility, indecomposability, and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity. The present paper is devoted to a detailed investigation of these invariants. The results give a complete characterization of irreducible snarks and characterizations of k-simple snarks for k ≤ 6. A number of problems that have arisen in this research conclude the paper. © 1996 John Wiley & Sons, Inc.     

On Coloring Problems of Snark Families

Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Theorem. In 2003, Cavicchioli et al. proved that for snarks with less than 30 vertices, the total chromatic number is 4, and proposed the problem of finding (if any) the smallest snark which is not 4-total colorable. Since then, only two families of snarks have had their total chromatic number determined to be 4, namely the Flower Snark family and the Goldberg family.We prove that the total chromatic number of the Loupekhine family is 4. We study the dot product, a known operation to construct snarks. We consider families of snarks using the dot product, particularly subfamilies of the Blanusa families, and obtain a 4-total coloring for each family. We study edge coloring properties of girth trivial snarks that cannot be extended to total coloring. We classify the snark recognition problem as CoNP-complete and establish that the chromatic number of a snark is 3.     

Non-Bicritical Critical Snarks

 Snarks are cubic graphs with chromatic index χ^′=4. A snark G is called critical if χ^′ (G−{v,w})=3 for any two adjacent vertices v and w, and it is called bicritical if χ^′ (G−{v,w})=3 for any two vertices v and w. We construct infinite families of critical snarks which are not bicritical. This solves a problem stated by Nedela and Škoviera. Revised: January 11, 1999     

Snarks and Flow-Critical Graphs

It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutteʼs Flow Conjectures.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

A result on the total colouring of powers of cycles

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. The Total Colouring Conjecture (TCC) states that for every simple graph G, χ_T(G)?Δ(G)+2. This work verifies the TCC for powers of cycles even and 2<k<n/2, showing that there exists and can be polynomially constructed a (Δ(G)+2)-total colouring for these graphs.     

Altitude of regular graphs with girth at least five

The altitude of a graph G is the largest integer k such that for each linear ordering f of its edges, G has a (simple) path P of length k for which f increases along the edge sequence of P. We determine a necessary and sufficient condition for cubic graphs with girth at least five to have altitude three and show that for r?4, r-regular graphs with girth at least five have altitude at least four. Using this result we show that some snarks, including all but one of the Blanus?a type snarks, have altitude three while others, including the flower snarks, have altitude four. We construct an infinite class of 4-regular graphs with altitude four.     

Entire colouring of plane graphs

It was conjectured by Kronk and Mitchem in 1973 that simple plane graphs of maximum degree Δ are entirely (Δ+4)-colourable, i.e., the vertices, edges, and faces of a simple plane graph may be simultaneously coloured with Δ+4 colours in such a way that adjacent or incident elements are coloured by distinct colours. Before this paper, the conjecture has been confirmed for Δ?3 and Δ?6 (the proof for the Δ=6 case has a correctable error). In this paper, we settle the whole conjecture in the positive. We prove that if G is a plane graph with maximum degree 4 (parallel edges allowed), then G is entirely 8-colourable. If G is a plane graph with maximum degree 5 (parallel edges allowed), then G is entirely 9-colourable.     

Removable edges in cyclically 4-edge-connected cubic graphs

We consider various ways of obtaining smaller cyclically 4-edge-connected cubic graphs from a given such graph. In particular, we consider removable edges: an edgee of a cyclically 4-edge-connected cubic graphG is said to be removable ifG is also cyclically 4-edge-connected, whereG is the cubic graph obtained fromG by deletinge and suppressing the two vertices of degree 2 created by the deletion. We prove that any cyclically 4-edge-connected cubic graphG with at least 12 vertices has at least 1/5(|E(G)| + 12) removable edges, and we characterize the graphs with exactly 1/5(|E(G)| + 12) removable edges.This work was carried out while the first author held a Niels Bohr Fellowship from the Royal Danish Academy of Sciences.     

A PTAS for the sparsest 2-spanner of 4-connected planar triangulations

A t-spanner of an undirected, unweighted graph G is a spanning subgraph of G with the added property that for every pair of vertices in G, the distance between them in is at most t times the distance between them in G. We are interested in finding a sparsest t-spanner, i.e., a t-spanner with the minimum number of edges. In the general setting, this problem is known to be NP-hard for all t2. For t5, the problem remains NP-hard for planar graphs, whereas for t{2,3,4}, the complexity of this problem on planar graphs is still unknown. In this paper we present a polynomial time approximation scheme for the problem of finding a sparsest 2-spanner of a 4-connected planar triangulation.     

Distinguishing chromatic numbers of complements of Cartesian products of complete graphs

The distinguishing chromatic number of a graph, $G$, is the minimum number of colours required to properly colour the vertices of $G$ so that the only automorphism of $G$ that preserves colours is the identity. There are many classes of graphs for which the distinguishing chromatic number has been studied, including Cartesian products of complete graphs (Jerebic and Klav?ar, 2010). In this paper we determine the distinguishing chromatic number of the complement of the Cartesian product of complete graphs, providing an interesting class of graphs, some of which have distinguishing chromatic number equal to the chromatic number, and others for which the difference between the distinguishing chromatic number and chromatic number can be arbitrarily large.     

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 J _k , of order 4k and a Goldberg snark, say B _k , of order 8k. We determine the automorphism groups of J _k and B _k for every k and prove that they are isomorphic to the dihedral group D _4k of order 4k. Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project “Strutture Geometriche, Combinatoria e loro Applicazioni”.     

A survey on snarks and new results: Products,reducibility and a computer search

In this paper we survey recent results and problems of both theoretical and algorithmic character on the construction of snarks—non-trivial cubic graphs of class two, of cyclic edge-connectivity at least 4 and with girth ≥ 5. We next study the process, also considered by Cameron, Chetwynd, Watkins, Isaacs, Nedela, and Sˇkoviera, of splitting a snark into smaller snarks which compose it. This motivates an attempt to classify snarks by recognizing irreducible and prime snarks and proving that all snarks can be constructed from them. As a consequence of these splitting operations, it follows that any snark (other than the Petersen graph) of order ≤ 26 can be built as either a dot product or a square product of two smaller snarks. Using a new computer algorithm we have confirmed the computations of Brinkmann and Steffen on the classification of all snarks of order less than 30. Our results recover the well-known classification of snarks of order not exceeding 22. Finally, we prove that any snark G of order ≤ 26 is almost Hamiltonian, in the sense that G has at least one vertex v for which G \ v is Hamiltonian. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 57–86, 1998     

Extension of Turán's Theorem to the 2-Stability Number

 Given a graph G with n vertices and stability number α(G), Turán's Theorem gives a lower bound on the number of edges in G. Furthermore, Turán has proved that the lower bound is only attained if G is the union of α(G) disjoint balanced cliques. We prove a similar result for the 2-stability number α₂(G) of G, which is defined as the largest number of vertices in a 2-colorable subgraph of G. Given a graph G with n vertices and 2-stability number α₂(G), we give a lower bound on the number of edges in G and characterize the graphs for which this bound is attained. These graphs are the union of isolated vertices and disjoint balanced cliques. We then derive lower bounds on the 2-stability number, and finally discuss the extension of Turán's Theorem to the q-stability number, for q>2. Received: July 21, 1999 Final version received: August 22, 2000 Present address: GERAD, 3000 ch. de la Cote-Ste-Catherine, Montreal, Quebec H3T 2A7, Canada. e-mail: Alain.Hertz@gerad.ca     

A note on on-line ranking number of graphs

A k-ranking of a graph G = (V, E) is a mapping ϕ: V → {1, 2, ..., k} such that each path with end vertices of the same colour c contains an internal vertex with colour greater than c. The ranking number of a graph G is the smallest positive integer k admitting a k-ranking of G. In the on-line version of the problem, the vertices v ₁, v ², ..., v _n of G arrive one by one in an arbitrary order, and only the edges of the induced graph G[{v ₁, v ₂, ..., v _i }] are known when the colour for the vertex v _i has to be chosen. The on-line ranking number of a graph G is the smallest positive integer k such that there exists an algorithm that produces a k-ranking of G for an arbitrary input sequence of its vertices. We show that there are graphs with arbitrarily large difference and arbitrarily large ratio between the ranking number and the on-line ranking number. We also determine the on-line ranking number of complete n-partite graphs. The question of additivity and heredity is discussed as well.     

Disproof of a conjecture in graph reconstruction theory

In his thesis [3] B. D. Thatte conjectured that ifG=G ₁,G ₂,...G _n is a sequence of finitely many simple connected graphs (isomorphic graphs may occur in the sequence) with the same number of vertices and edges then their shuffled edge deck uniquely determines the graph sequence (up to a permutation). In this paper we prove that there are such sequences of graphs with the same shuffled edge deck.This research was partially supported by Hungarian National Foundation of Scientific Research Grant no. 1812     

Circular flow numbers of regular multigraphs

The circular flow number F_c(G) of a graph G = (V, E) is the minimum r ϵ ℚ such that G admits a flow ϕ with 1 ≤ ϕ (e) ≤ r − 1, for each e ϵ E. We determine the circular flow number of some regular multigraphs. In particular, we characterize the bipartite (2t+1)‐regular graphs (t ≥ 1). Our results imply that there are gaps for possible circular flow numbers for (2t+1)‐regular graphs, e.g., there is no cubic graph G with 3 < F_c(G) < 4. We further show that there are snarks with circular flow number arbitrarily close to 4, answering a question of X. Zhu. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 24–34, 2001     

Cubic graphs without cut‐vertices having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all possible paths in G. The entry (i,j) of this matrix is the number of paths in G having initial vertex i and length j. It is known that there are cubic graphs on 62 vertices having the same path layer matrix (A. A. Dobrynin. J Graph Theory 17 (1993) 1–4). A new upper bound of 36 vertices for the least order of such cubic graphs is established. This bound is realized by cubic graphs without cut‐vertices. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 177–182, 2001     

On the harmonious coloring of collections of graphs

The hermonious coloring number of the graph G, HC(G), is the smallest number of colors needed to label the vertices of G such that adjacent vertices received different colors and no two edges are incident with the same color pair. In this paper, we investigate the HC-number of collections of disjoint paths, cycles, complete graphs, and complete bipartite graphs. We determine exact expressions for the HC-number of collections of paths and 4m-cycles. © 1995, John Wiley & Sons, Inc.     

Looseness of Plane Graphs

A face of a vertex coloured plane graph is called loose if the number of colours used on its vertices is at least three. The looseness of a plane graph G is the minimum k such that any surjective k-colouring involves a loose face. In this paper we prove that the looseness of a connected plane graph G equals the maximum number of vertex disjoint cycles in the dual graph G* increased by 2. We also show upper bounds on the looseness of graphs based on the number of vertices, the edge connectivity, and the girth of the dual graphs. These bounds improve the result of Negami for the looseness of plane triangulations. We also present infinite classes of graphs where the equalities are attained.