Graph Coloring Problems

“…We have seen that the ds-tree T [ ] is a tree rooted at r such that: {r; 1 ; 2 ; : : : ; n } is its vertex set and k ; i is an edge if and only (1 6 i 6 n) is deÿned to be a rooted tree such that: (i) i is the root of the tree; (ii) p is a node i i ¿ p and p in ( i+1 ; i+2 ; : : : ; n ); that is, i + 1 6 p 6 n; Fig. 5 illustrates the way we can construct the ds-trees T [7]; T [9] and T [3] of the permutation , where is the same permutation used in this paper.…”

Section: The Decreasing-subsequence Treesmentioning

confidence: 99%

“…In Fig. 7(b), the pair (1; 3) is a link-pair in the trees T [12]; T [7]; T [11]; T [8]; T [6]; T [10] and T [9], while it is not a link-pair in the trees T [2]; T [4]; T [3] and T [1].…”

Section: The Decreasing-subsequence Treesmentioning

confidence: 99%

Coloring permutation graphs in parallel

Nikolopoulos

1999

Electronic Notes in Discrete Mathematics

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A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having speciÿc key properties. We propose an e cient parallel algorithm which colors an n-node permutation graph in O(log 2 n) time using O(n 2 =log n) processors on the CREW PRAM model. Speciÿcally, given a permutation we construct a tree T * [ ], which we call coloring-permutation tree, using certain combinatorial properties of . We show that the problem of coloring a permutation graph is equivalent to ÿnding vertex levels in the coloring-permutation tree. ?

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“…Tait proved in 1880 (see Jensen and Toft [3] page 16) that there is a one-to-one correspondance between the 4-vertex-colorings of a planar triangulation and the proper 3-edge-colorings of the dual 1 graph. A proper 3-edge coloring of the dual graph can also be expressed as an edge-coloring of the triangulation such that every triangle receives all three colors, or equivalently, the union of any two colors is a spanning quadrangulation.…”

Section: Introductionmentioning

confidence: 99%

Spanning quadrangulations of triangulated surfaces

Kündgen

Thomassen

2017

Abh. Math. Semin. Univ. Hambg.

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In this paper we study alternating cycles in graphs embedded in a surface. We observe that 4-vertex-colorability of a triangulation on a surface can be expressed in terms of spanninq quadrangulations, and we establish connections between spanning quadrangulations and cycles in the dual graph which are noncontractible and alternating with respect to a perfect matching. We show that the dual graph of an Eulerian triangulation of an orientable surface other than the sphere has a perfect matching M and an M -alternating noncontractible cycle. As a consequence, every Eulerian triangulation of the torus has a nonbipartite spanning quadrangulation. For an Eulerian triangulation G of the projective plane the situation is different: If the dual graph G * is nonbipartite, then G * has no noncontractible alternating cycle, and all spanning quadrangulations of G are bipartite. If the dual graph G * is bipartite, then it has a noncontractible, M -alternating cycle for some (and hence any) perfect matching, G has a bipartite spanning quadrangulation and also a nonbipartite spanning quadrangulation.

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Graph Coloring Problems

Cited by 630 publications

References 0 publications

The concentration of the chromatic number of random graphs

The concentration of the chromatic number of random graphs

Coloring permutation graphs in parallel

Spanning quadrangulations of triangulated surfaces

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