Showing papers on "Edge coloring published in 1982"
TL;DR: Coloring algorithms that run in time $O(\min (m(\log n)^2 ,n^2 \log n))$ are presented and can be used to find maximum cardinality matchings on regular bipartite graphs in the above time bound.
Abstract: A minimum edge coloring of a bipartite graph is a partition of the edges into $\Delta $ matchings, where $\Delta $ is the maximum degree in the graph. Coloring algorithms that run in time $O(\min (m(\log n)^2 ,n^2 \log n))$ are presented. The algorithms rely on an efficient procedure for the special case of $\Delta $ an exact power of two. The coloring algorithms can be used to find maximum cardinality matchings on regular bipartite graphs in the above time bound. An algorithm for coloring multigraphs with large multiplicities is also presented.
159 citations
TL;DR: This article proves and applies to examples three types of general theorem which have no analogs for ordinary graph coloring and proves a balanced expansion theorem which reduces calculation of the chromatic and Whitney polynomials to that of the simpler balanced polynmials.
55 citations
TL;DR: The following theorem is proved: If G is a simple graph and @m is the maximal cardinality of a triangle-free edge set of G, then the edges of G can be covered by @m triangles and edges.
29 citations
5 May 1982
TL;DR: A simple and efficient coloring algorithm which guarantees-&-Agrave; (G)-&-le;x(G)n (equation), a considerable improve over the current bounds is presented.
Abstract: Let A be a graph coloring algorithm. Denote by -&-Agrave; (G) the ratio between the maximum number of colors A will use to color the graph G, and the chromatic number of G,x(G). For most existing polynomial coloring algorithms, -&-Agrave;(G) can be as bad as O(n), where n is the number of vertices in G. The best currently known algorithm guarantees -&-Agrave; (G)-&-equil;O(n/logn). In this paper we present a simple and efficient coloring algorithm which guarantees -&-Agrave;(G)-&-le;x(G)n (equation), a considerable improvem-&-edot;nt over the current bounds.
25 citations
TL;DR: The generalized Petersen graph P(6k + 3, 2) has exactly 3 Hamiltonian cycles for k ≥ 0, but for k ≤ 2 is not uniquely edge colorable, which disproves a conjecture of Greenwell and Kronk.
Abstract: The generalized Petersen graph P(6k + 3, 2) has exactly 3 Hamiltonian cycles for k ≥ 0, but for k ≥ 2 is not uniquely edge colorable. This disproves a conjecture of Greenwell and Kronk [1].
20 citations
TL;DR: The concept of k -tuple coloring the nodes of a graph is extended to specify that adjacent nodes must have i colors in common, and results for a minimum such coloring are obtained for bipartite graphs and odd cycles.
13 citations
TL;DR: It is pointed out that the edge coloring approach results in the most efficient algorithms for the decomposition problem.
Abstract: A counterexample is given to show the partial correctness of a matrix decomposition algorithm proposed by Ramanujam in 1973. It is pointed out that the edge coloring approach results in the most efficient algorithms for the decomposition problem.
12 citations
TL;DR: In this note, it is shown that there is no snark (i.e. a cyclically 4-edge connected cubic graph, without square, and with chromatic index 4) with exactly 16 vertices.
10 citations
TL;DR: It is found that this heuristic graph coloring algorithm is effective for all n less than N, where InInN is about the same as np(n).
Abstract: In this paper a heuristic graph coloring algorithm as applied to random graphs is analyzed theoretically. The result obtained by expanding on the results of [4] is presented first: let x1(n, cnδ−1) be the random variable representing a solution obtained by applying the algorithm to a random n-vertex graph having edges with probability p(n)=cnδ− (where c is a constant and 1/2≪δ≦1). Also let x1(n, cnδ−1) be the random variable representing the chromatic number of the graph. Then x1(n, cnδ−1)/x*(n, cnδ−1)≦(1+ϵ)2δ/(2δ−1)(pr.) holds. Here ϵ represents a value sufficiently smaller than 1. By contrast, it is important from the practical point of view to find an algorithm that is effective for an arbitrary random n-vertex graph. In this paper the above-mentioned algorithm is evaluated using models with p(n)≪ It is found that this algorithm is effective for all n less than N, where InInN is about the same as np(n).
3 citations
TL;DR: In this article, it was shown that for any formula p of the first-order predicate calculus, one can construct a finite set Bp of tinted graphs of order 3 such that ⌉⌉p if and only if there exists a Bp-regular tinted graph not admitting a Cp regular coloring.
Abstract: A tinted graph is a graph whose arcs are colored with certain colors. A colored graph is a graph whose vertices are colored with certain colors. If M is the set of tinted (or colored tinted) graphs of order k and G is a tinted (or colored tinted) graph, then we shall say that G is M-regular (or M-regularly colored) if all its subgraphs of order k belong to M. We shall show how, for any formula p of the first-order predicate calculus, to construct a finite set Bp of tinted graphs of order 3 and a finite set Cp of colored tinted graphs of order 2 such that ¦-p if and only if there exists a Bp-regular tinted graph not admitting a Cp-regular coloring. Hadwiger's conjecture (HC) is as follows: If no subgraph of a graph without loops G is contractible to a complete graph of order n, then the vertices of G can be colored in n−1 colors such that neighboring vertices are colored with different colors. We construct a formula X of the first-order predicate calculus such that HC is equivalent with ⌉⊢X. Thus, HC reduces to HC1: if all subgraphs of order 3 of the tinted graph G belong to BX, then G is CX-regularly colorable. Here BX and CX are specific finite sets of tinted graphs of order 3 and colored tinted graphs of order 2, respectively.
TL;DR: The number of colors needed to colour the blocks of a cyclic Steiner 2-design S(2, k, v) is at most v as discussed by the authors, where v is the number of colours needed to color the blocks.
Abstract: The number of colors needed to colour the blocks of a cyclic Steiner 2- design S(2, k, v) is at most v.
TL;DR: The purpose of this note is to describe a method for computing the chromatic index of STS, and to describe results obtained in testing this method on small STS.
Abstract: A Steiner triple system of order v, denoted STS (v), is a pair (V, B); V is a u-set of elements and B is a collection of 3-subsets of V called blocks. Each unordered pair of elements is contained in precisely one block. There is a substantial body of research on STS, partially because of their wide applicability in the design of experiments, and in the theory of error-correcting codes. In the design of statistical experiments, each block corresponds to a 'test' of the three elements in the block. In this context, we can view disjoint blocks as tests which can be carried out simultaneously. In terms of STS, we define a colour class to be a set of pairwise disjoint blocks. A k-block colouring is a partition of the block set into k colour classes; the chromatic index is the least k for which such a colouring exists. In our example, the chromatic index is precisely the time required for the entire experiment. In other applications of STS, similar reasons exist for studying the chromatic index. The chromatic index of STS is also a problem of some interest in combinatorics as a restriction of certain investigations of set systems. In computer science, further motivation arises from the substantial interest in the chromatic index of graphs, and applications in scheduling (see Ref. 5 and references therein). The purpose of this note is to describe a method for computing the chromatic index of STS, and to describe results obtained in testing this method on small STS.
TL;DR: The generalization to voltage graphs is outlined, where the chromatic polynomial with the same enumerative and algebraic properties as for ordinary graphs is generalized to signed graphs.