Results 1  10
of
14
Decompositions And Reductions Of Snarks
, 1996
"... . According to M. Gardner [9], a snark is a nontrivial cubic graph whose edges cannot be properly coloured by three colours. 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 th ..."
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. According to M. Gardner [9], a snark is a nontrivial cubic graph whose edges cannot be properly coloured by three colours. 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 kedgecut producing components G 1 and G 2 , then either one of G 1 and G 2 is not 3edgecolourable, or by adding a `small' number of vertices to either component one can obtain snarks ~ G 1 and ~ G 2 whose order does not exceed that of G. The two situations lead to a definition of a kreduction and kdecomposition of G. Snarks that for m ! k do not admit mreductions, mdecompositions or both are kirreducible, kindecomposable and ksimple, respectively. The irreducibility, indecomposability and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity. The present paper...
Chromatic Index Critical Graphs of Orders 11 and 12
, 1997
"... A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results thi ..."
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A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three nontrivial chromaticindex critical graphs of order 12. 1 Introduction A famous theorem of Vizing [20] states that the chromatic index Ø 0 (G) of a simple graph G is \Delta(G) or \Delta(G) + 1, where \Delta(G) denotes the maximum vertex degree in G. A graph G is class 1 if Ø 0 (G) = \Delta(G) and it is class 2 otherwise. A class 2 graph G is (chromatic index) critical if Ø 0 (G \Gamma e) ! Ø 0 (G) for each edge e of G. If we want to stress the maximum vertex degree of a critical graph G we say G is \Delta(G)critical. Critical graphs of odd order are easy to construct while not much is known about critical graphs of even order. One reas...
Factorisation of Snarks
, 2010
"... We develop a theory of factorisation of snarks — cubic graphs with edgechromatic number 4 — based on the classical concept of the dot product. Our main concern are irreducible snarks, those where the removal of every nontrivial edgecut yields a 3edgecolourable graph. We show that if an irreducib ..."
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We develop a theory of factorisation of snarks — cubic graphs with edgechromatic number 4 — based on the classical concept of the dot product. Our main concern are irreducible snarks, those where the removal of every nontrivial edgecut yields a 3edgecolourable graph. We show that if an irreducible snark can be expressed as a dot product of two smaller snarks, then both of them are irreducible. This result constitutes the first step towards the proof of the following “uniquefactorisation” theorem: Every irreducible snark G can be factorised into a collection {H1,...,Hn} of cyclically 5connected irreducible snarks such that G can be reconstructed from them by iterated dot products. Moreover, such a collection is unique up to isomorphism and ordering of the factors regardless of the way in which the decomposition was performed. The result is best possible in the sense that it fails for snarks that are close to being irreducible but themselves are not irreducible. Besides this theorem, a number of other results are proved. For example, the uniquefactorisation theorem is extended to the case of factorisation with respect to a preassigned subgraph K which is required to stay intact during the whole factorisation process. We show that if K has order at least 3, then the theorem holds, but is false when K has order 2.
Theorems and Computations . . .
, 2007
"... The circular chromatic number provides a more refined measure of colourability of graphs, than does the ordinary chromatic number. Thus circular colouring is of substantial importance wherever graph colouring is studied or applied, for example, to scheduling problems of periodic nature. Precisely, ..."
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The circular chromatic number provides a more refined measure of colourability of graphs, than does the ordinary chromatic number. Thus circular colouring is of substantial importance wherever graph colouring is studied or applied, for example, to scheduling problems of periodic nature. Precisely, the circular chromatic number of a graph G, denoted by χc(G), is the smallest ratio p/q of positive integers p and q for which there exists a mapping c: V (G) → {1,2,...,p} such that q � c(u) − c(v)  � p − q for every edge uv of G. We present some known and new results regarding the computation of the circular chromatic number. In particular, we prove a lemma which can be used to improve the ratio of some circular colourings. These results are later used to bound the circular chromatic number of the plane unitdistance graph, the projective plane orthogonality graph, generalized Petersen graphs, and squares of graphs. Some of the computations in this thesis are computer assisted. Neˇsetˇril’s “pentagon problem”, asks whether the circular chromatic number of every cubic graph having sufficiently high girth is at most 5/2. We prove that the statement of the
The Circular Chromatic Index of Goldberg
"... We determine the exact values of the circular chromatic index of the Goldberg snarks, and of a related family, the twisted Goldberg snarks. Key words: Edge colouring, Circular colouring, Snark, Goldberg snarks 1 ..."
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We determine the exact values of the circular chromatic index of the Goldberg snarks, and of a related family, the twisted Goldberg snarks. Key words: Edge colouring, Circular colouring, Snark, Goldberg snarks 1
Nonbicritical Critical Snarks
"... Snarks are cubic graphs with chromatic index Ø 0 = 4. A snark G is called critical if Ø 0 (G \Gamma fv; wg) = 3 for any two adjacent vertices v and w, and it is called bicritical if Ø 0 (G \Gamma fv; wg) = 3 for any two vertices v and w. We construct infinite families of critical snarks which ..."
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Snarks are cubic graphs with chromatic index Ø 0 = 4. A snark G is called critical if Ø 0 (G \Gamma fv; wg) = 3 for any two adjacent vertices v and w, and it is called bicritical if Ø 0 (G \Gamma fv; wg) = 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 Skoviera in [7]. 1 Introduction We are using standard graph theoretical terminology and notation in this paper. We define a snark to be a cubic graph with chromatic index Ø 0 = 4. The study of reduction of snarks is as old as the study of snarks itself. The first reductions which are considered were edgecut reduction and decomposition of snarks, cf. [2, 5, 6, 7, 8, 11, 12]. In [7] Nedela and Skoviera gave these two concepts a precise meaning, and we refer the reader to this paper for a more extensive introduction. In that paper the reduction of snarks is defined as follows. Let G = (V (G); E(G)) be a snark and let F ae E(G)...
Classifications and Characterisations of Snarks
, 1994
"... This paper considers vertexreductions of snarks. Each reduction naturally divides the class of snarks into three classes. The first class contains those snarks which are not reducible to a snark. The second contains those which are not reducible to a 3colorable graph and the third one consists of ..."
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This paper considers vertexreductions of snarks. Each reduction naturally divides the class of snarks into three classes. The first class contains those snarks which are not reducible to a snark. The second contains those which are not reducible to a 3colorable graph and the third one consists of the snarks which are reducible to a snark as well as to a 3colorable graph. We characterise all these classes in terms of 2factors. Furthermore we characterise snarks which are divided into 3colorable components by the removal of any edge cutset in terms of 3critical subgraphs and show that there are infinitely many such snarks. 1 Introduction We follow the terminology and notation of Bondy and Murty [2]. We consider proper edge colorings of 3regular graphs. The set of 3regular graphs is divided into two classes by a famous theorem of Vizing which states that each simple loopless graph is edgecolorable with \Delta or \Delta +1 colors, see e.g. [2]. Class I contains the 3colorable gr...
On polyhedral embeddings of cubic graphs
, 2004
"... Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is veri ..."
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Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every nonorientable surface S, there exists a non 3edgecolorable graph, which polyhedrally embeds in S.