Decompositions and reductions of snarks

Abstract

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.

Irreducible snarks of given order and cyclic connectivity

“…It is therefore natural to ask which snarks are to be considered "nontrivial", so that all other snarks could be derived from them by certain simple operations. The problem of nontriviality is thus essential for understanding the structure of snarks and has been discussed by many authors, see for example [3,11,15,17].…”

“…As shown by Cameron et al [3], these restrictions still leave snarks which arise from smaller snarks by certain operations such as the dot-product. This fact suggests that the two standard nontriviality conditions can be somewhat arbitrary which calls for a more systematic approach to the phenomenon of nontriviality of snarks (see [17,15]). …”

“…In this paper we follow the approach of Nedela and Škoviera [15] which is based on the concepts of reductions and decompositions of snarks. For this approach, it is convenient to allow cubic graphs to have both parallel edges and loops, and to leave the definition of a snark as broad as possible.…”

“…In [15], Nedela and Škoviera characterized k-irreducible snarks in terms of cyclic connectivity and criticality. Recall that a graph is cyclically k-connected if it does not contain a cycle separating edge-cut of cardinality smaller than k. The cyclic connectivity of a graph is the smallest number k for which the graph is cyclically k-connected (apart from three small graphs: K 4 , K 3,3 , and the threefold K 2 for which cyclic connectivity is defined to be the cycle rank).…”

On Hypohamiltonian and Almost Hypohamiltonian Graphs

“…the graph itself is not 3-edge-colourable but the removal of any two distinct vertices yields a 3-edge-colourable graph. Nedela andŠkoviera [101] showed that every cubic bicritical graph is cyclically 4-edge-connected and has girth at least 5. Therefore, every cubic hypohamiltonian graph with chromatic index 4 must be a snark.…”