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11
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 the dis ..."
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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...
ChromaticIndex Critical Graphs of Even Order
, 1997
"... A kcritrical graph G has maximum degree k 0, chromatic index Ø 0 (G) = k + 1 and Ø 0 (G \Gamma e) ! k + 1 for each edge e of G. The Critical Graph Conjecture, Jakobsen [8] and Beineke, Wilson [1], claims that every kcritical graph is of odd order. Fiorini and Wilson [6] conjectured that ev ..."
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A kcritrical graph G has maximum degree k 0, chromatic index Ø 0 (G) = k + 1 and Ø 0 (G \Gamma e) ! k + 1 for each edge e of G. The Critical Graph Conjecture, Jakobsen [8] and Beineke, Wilson [1], claims that every kcritical graph is of odd order. Fiorini and Wilson [6] conjectured that every kcritical graph of even order has a 1factor. Chetwynd and Yap [4] stated the problem whether it is true that if G is a kcritical graph of odd order, then G \Gamma v has a 1factor for every vertex v of minimum degree. These conjectures are disproved and the problem is answered in the negative for k 2 f3; 4g. We disprove these conjectures and answer the problem in the negative for all k 3. We also construct kcritical graphs on n vertices with degree sequence 23 2 4 n\Gamma3 , answering a question of Yap [11]. 1 Introduction We consider connected multigraphs M = (V (M); E(M)) without loops, where V (M) (E(M)) denotes the set of vertices (edges) of M . The degree dM (v) of a v...
Cyclically 5Edge Connected NonBicritical Critical Snarks
, 1997
"... Snarks are bridgeless 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. For any k 2 we construct cyclically 5edge connected critical snarks G having an independent set I of at least k vertices such that ..."
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Snarks are bridgeless 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. For any k 2 we construct cyclically 5edge connected critical snarks G having an independent set I of at least k vertices such that Ø 0 (G \Gamma I) = 4. For k = 2 this solves a problem of Nedela and Skoviera [5]. 1 Introduction A snark is a bridgeless cubic graph with chromatic index Ø 0 = 4. The study of the reduction of snarks is as old as the study of these graphs in itself. For a detailed introduction to this topic we refer the reader to one of [2, 4, 5, 6, 8, 11, 12]. This note deals with a reduction of snarks introduced by Nedela and Skoviera in [5]. Let G = (V (G); E(G)) be a snark and let F ae E(G) be a kedge cut (k 0) whose removal divides G into two components H 1 and H 2 . If the chromatic index of one of the components is 4, say Ø 0 (H 1 ) = 4, then H 1 can be extended to a snark H = (V (H); E(...
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
ON FULKERSON CONJECTURE J.L. FOUQUET AND J.M. VANHERPE
, 906
"... Abstract. If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect ..."
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Abstract. If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FRtriple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FRtriples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks. 1.
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.