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31
Excluding Minors in Cubic Graphs
 Combin. Probab. Comput
, 1996
"... . Let P 10 ne be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P 10 ne. The decomposition is used to show that graphs in this class are 3edgecolourable. We also consider an application to a conjecture du ..."
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. Let P 10 ne be the graph obtained by deleting an edge from the Petersen graph. We give a decomposition theorem for cubic graphs with no minor isomorphic to P 10 ne. The decomposition is used to show that graphs in this class are 3edgecolourable. We also consider an application to a conjecture due to Grotzsch which states that a planar graph is 3edgecolourable if and only if it is fractionally 3edgecolourable. 1985 Mathematics Subject Classification: 05C50,05C75. Key Words and Phrases: Planar graph, Petersen Graph, Four Colour Problem, cubic, matching polyhedron, integer decomposition property, edgecolouring. 1 Background We consider loopless graphs G = (V; E) with node set V and edge set E. For an edge e 2 E, we denote by G=e the graph obtained by contracting the edge e, i.e., identifying its ends and deleting the resulting loop. A subgraph (respectively induced subgraph) of G is a graph obtained by deleting nodes or edges (respectively deleting nodes) of G. Informally, a...
COMBINATORIAL STACKS AND THE FOURCOLOUR THEOREM
, 2005
"... Abstract. We interpret the number of good fourcolourings of the faces of a trivalent, spherical polyhedron as the 2holonomy of the 2connection of a fibered category, ϕ, modeled on Rep f (sl2) and defined over the dual triangulation, T. We also build an sl2bundle with connection over T, that is a ..."
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Abstract. We interpret the number of good fourcolourings of the faces of a trivalent, spherical polyhedron as the 2holonomy of the 2connection of a fibered category, ϕ, modeled on Rep f (sl2) and defined over the dual triangulation, T. We also build an sl2bundle with connection over T, that is a global, equivariant section of ϕ, and we prove that the fourcolour theorem is equivalent to the fact that the connection of this sl2bundle vanishes nowhere. This interpretation is proposed as a first step toward a cohomological proof of the fourcolour theorem. Contents
Algorithms for the Total Colorings of Graphs
, 1993
"... This thesis presents efficient algorithms for finding total colorings of graphs. A total coloring of a graph G is a coloring of all vertices and edges of G such that any two adjacentvertices receive different colors, anytwo adjacent edges receive different colors, and any edge receives a color diffe ..."
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This thesis presents efficient algorithms for finding total colorings of graphs. A total coloring of a graph G is a coloring of all vertices and edges of G such that any two adjacentvertices receive different colors, anytwo adjacent edges receive different colors, and any edge receives a color different from the colors of its two ends. The total coloring problem asks to find a total coloring of a given graph with the minimum number of colors. This problem is NPhard and hence it is very unlikely that there is an efficient algorithm to solve the total coloring problem. On the other hand, it is known that many NPhard problems, including the ordinary vertex and edgecoloring problems, can be efficiently solved for some restricted classes of graphs such as partial ktrees, seriesparallel graphs, degenerate graphs, and so on.
An Algorithm for Cyclic Edge Connectivity of Cubic Graphs
, 2003
"... The cyclic edge connectivity is the size of the smallest edge cut in a graph such that at least two of the parts of the graph are not acyclic. We present an algorithm running in time O(n n) for computing the cyclic edge connectivity of nvertex cubic graphs. ..."
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The cyclic edge connectivity is the size of the smallest edge cut in a graph such that at least two of the parts of the graph are not acyclic. We present an algorithm running in time O(n n) for computing the cyclic edge connectivity of nvertex cubic graphs.
Bounding the circumference of 3connected clawfree graphs
"... The circumference of a graph is the length of its longest cycles. A result of Jackson and Wormald implies that the circumference of a 3connected clawfree graph is at least 1 2 nlog 150 2. In this paper we improve this lower bound to Ω(n log 3 2), and our proof implies a polynomial time algorithm f ..."
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The circumference of a graph is the length of its longest cycles. A result of Jackson and Wormald implies that the circumference of a 3connected clawfree graph is at least 1 2 nlog 150 2. In this paper we improve this lower bound to Ω(n log 3 2), and our proof implies a polynomial time algorithm for finding a cycle of such length. Bondy and Simonovits showed that Θ(n log 9 8) is an upper bound.
OPTIMIZING PHYLOGENETIC DIVERSITY ACROSS TWO TREES
"... Abstract. We present a polynomialtime algorithm for finding an optimal set of taxa that maximizes the weightedsum of the phylogenetic diversity across two phylogenetic trees. This resolves one of the challenges proposed as part of the Phylogenetics Programme held at the Isaac Newton Institute for ..."
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Abstract. We present a polynomialtime algorithm for finding an optimal set of taxa that maximizes the weightedsum of the phylogenetic diversity across two phylogenetic trees. This resolves one of the challenges proposed as part of the Phylogenetics Programme held at the Isaac Newton Institute for Mathematical Sciences (Cambridge, 2007). It also completely closes the gap between optimizing phylogenetic diversity on one tree, which is known to be in P, and optimizing phylogenetic diversity across three or more trees, which is known to be NPhard. 1.
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