Showing papers on "Edge coloring published in 1986"
1 Sep 1986
TL;DR: In this paper, the authors consider a special kind of multigraph, called a star-multigraph, which contains a vertex v*, called the star-centre, which is incident with each non-simple edge.
Abstract: The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map o: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G, where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.
103 citations
TL;DR: This paper presents an algorithm to find an edge coloring of a multigraph that never uses more than ⌊ 9 8 χ′ + 3 4 ⌋ colors and runs in O(|E|(|V| + Δ)) time, where E is the set of edges, and V is theSet of vertices.
54 citations
17 Jun 1986
TL;DR: A linear time algorithm to determine whether an arbitrary graph is outerplanar is described, which works without splitting the graph into its biconnected components or using bucket sort to give the adjacency lists a special order.
Abstract: This paper describes a linear time algorithm to determine whether an arbitrary graph is outerplanar. The algorithm uses an edge coloring technique and deletes successively vertices of degree less than or equal to two. If the degree of a vertex is two, both neighbors of the vertex are joined by an edge. The algorithm works without splitting the graph into its biconnected components or using bucket sort to give the adjacency lists a special order.
43 citations
TL;DR: In this article, the authors give an efficient algorithm to color the vertices of an outerplanar graph with the minimum number of colors, based on systematic coloring of elements (vertices and edges, respectively) of adjacent faces.
Abstract: The problems of finding values of the chromatic number and the chromatic index of a graph are NP-hard even for some restricted classes of graphs. Every outerplanar graph has an associated tree structure which facilitates algorithmic treatment. Using that structure, we give an efficient algorithm to color the vertices of an outerplanar graph with the minimum number of colors. We also establish algorithmically the value of the chromatic index of an outerplanar graph. Our algorithms are based on systematic coloring of elements (vertices and edges, respectively) of adjacent faces.
32 citations
TL;DR: Seymour's conjecture can be reduced to a conjecture about critical nonseparable graphs (in the sense of matching theory) and the latter conjecture is verified in the case of outerplanar graphs, thus proving that Seymour's conjecture holds for outerplanars graphs.
19 citations
TL;DR: There is a relationship between graph coloring and monotone functions defined on posets that permits us to deduce certain properties of the chromatic polynomial of a graph.
17 citations
TL;DR: An O(Δ|E|) algorithm is given, where Δ is the largest vertex degree, for finding an edge-maximal subgraph with a TR-formative edge-coloring in an arbitrary graph G, which can be used to construct improved implicit enumeration procedures for finding a maximum-weight clique in an arbitrarily graph.
15 citations
TL;DR: In this paper, the authors give upper bounds for the chromatic number (3, G) for bipartite (planar) graphs and generalize a result of S. Antonucci giving a lower bound for (2, G).
Abstract: The chromatic number relative to distance p, denoted by (p, G) is the minimum number of colors sufficing for coloring the vertices of G in such a way that any two vertices of distance not greater than p have distinct colors. We give upper bounds for the chromatic number (3, G) for bipartite (planar) graphs and generalize a result of S. Antonucci giving a lower bound for (2, G).
13 citations
TL;DR: The chromatic index of any multigraph which contains a vertex whose detetion results in a bipartite multigraph is determined.
9 citations
TL;DR: A decomposition scheme for coloring perfect graphs is described, which suggests that, under appropriate definition, highly connected perfect graphs might possess certain regular properties that are amenable to coloring algorithms.
Abstract: This paper describes a decomposition scheme for coloring perfect graphs. Based on this scheme, one need only concentrate on coloring highly connected (at least 3-connected) perfect graphs. This idea is illustrated on planar perfect graphs, which yields a straightforward coloring algorithm. We suspect that, under appropriate definition, highly connected perfect graphs might possess certain regular properties that are amenable to coloring algorithms.
7 citations
TL;DR: In this paper, it was shown that for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r − s for all e ∈ E(G/S).
Abstract: Let K denote the complete graph K2n+1 with each edge replicated r times and let χ′(G) denote the chromatic index of a multigraph G. A multigraph G is critical if χ′(G) > χ′(G/e) for each edge e of G. Let S be a set of sn – 1 edges of K. We show that, for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r – s for all e ∈ E(G/S). Plantholt [M. Plantholt, The chromatic index of graphs with a spanning star. J. Graph Theory 5 (1981) 5–13] proved this result in the case when r = 1.
TL;DR: In this article, a look-ahead rule is applied to determine a most suitable color for the vertices of a graph, and the influence of the vertex order at the beginning of the Sv algorithm (Lf or Sl ) is investigated.
TL;DR: Bounds are given on the number of colors required to color the edges of a graph (multigraph) such that each color appears at each vertex v at most m(ν) times.
Abstract: Bounds are given on the number of colors required to color the edges of a graph (multigraph) such that each color appears at each vertex v at most m(ν) times. The known results and proofs generalize in natural ways. Certain new edge-coloring problems, which have no counterparts when m(ν) = 1 for all ν ϵ V, are studied.
1 Jun 1986
TL;DR: The complexity of the class of generalized coloring problems, denoted GCP F,k , which arise in resource allocation and VLSI theory, ranges from polynomial to ∑ 2 p -complete, which represents the first natural graph problems to be complete for any intermediate slot of thePolynomial hyerarchy.
Abstract: We characterize the complexity of the class of generalized coloring problems, denoted GCP F,k , which arise in resource allocation and VLSI theory. Depending on the parameters, this complexity ranges from polynomial to ∑ 2 p -complete. The latter represent apparenly the first natural graph problems to be complete for any intermediate slot of the polynomial hyerarchy. A parallelizable algorithm for the polynomial problems is presented.
TL;DR: The chromatic number of G, Chr( G )⩽ r 2 r log 2 log 2 m (and this value is the best possible in a certain sense) is considered, which is a good coloring with m colors which uses at most r colors in the neighborhood of every vertex.