Showing papers in "Journal of Graph Theory in 1996"
TL;DR: In this article, it was shown that if s and t are non-negative integers, and if G is a graph with minimum degree s + t + 1, then the vertex set of G can be partitioned into two sets which induce subgraphs of minimum degree at least s and T, respectively.
Abstract: We prove a conjecture of C Thomassen: If s and t are non-negative integers, and if G is a graph with minimum degree s + t + 1, then the vertex set of G can be partitioned into two sets which induce subgraphs of minimum degree at least s and t, respectively © 1996 John Wiley & Sons, Inc
95 citations
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TL;DR: In this article, an efficient algorithm to generate regular cubic graphs with small vertex valency is presented, and the running times of a program based on this algorithm and designed to generate cubic graphs are below two natural benchmarks: (a) if N(n) denotes the number of pairwise non-isomorphic cubic objects with n vertices and T (n) the time needed for generating the list of all these objects, then T(n)/N(n)) decreases gradually for the observed values of n.
Abstract: In this paper an efficient algorithm to generate regular graphs with small vertex valency is presented. The running times of a program based on this algorithm and designed to generate cubic graphs are below two natural benchmarks: (a) If N(n) denotes the number of pairwise non-isomorphic cubic graphs with n vertices and T(n) the time needed for generating the list of all these graphs, then T(n)/N(n) decreases gradually for the observed values of n. This suggests that T(n)/N(n) might be uniformly bounded for all n, ignoring the time to write the outputs, but we are unable to prove this and in fact are not confident about it. (b) For programs that generate lists of non-isomorphic objects, but cannot a priori make sure to avoid the generation of isomorphic copies, the time needed to check a randomly ordered list of these objects for being non-isomorphic is a natural benchmark. Since for large lists (n ≥ 22, girth 3) existing graph isomorphism programs take longer to canonically label all of the N(n) graphs than our algorithm takes to generate them, our algorithm is probably faster than any method which does one or more isomorphism test for every graph. © 1996 John Wiley & Sons, Inc.
76 citations
TL;DR: In this paper, 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.
Abstract: According to M Gardner [“Mathematical Games: Snarks, Boojums, and Other Conjectures Related to the Four-Color-Map Theorem,” Scientific American, vol 234 (1976), pp 126–130], a snark is a nontrivial cubic graph whose edges cannot be properly colored by three colors 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 k-edge-cut producing components G1 and G2, then either one of G1 and G2 is not 3-edge-colorable, or by adding a “small” number of vertices to either component one can obtain snarks G1 and G2 whose order does not exceed that of G The two situations lead to a definition of a k-reduction and k-decomposition of G Snarks that for m < k do not admit m-reductions, m-decompositions, or both are k-irreducible, k-indecomposable, and k-simple, respectively The irreducibility, indecomposability, and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity The present paper is devoted to a detailed investigation of these invariants The results give a complete characterization of irreducible snarks and characterizations of k-simple snarks for k ≤ 6 A number of problems that have arisen in this research conclude the paper © 1996 John Wiley & Sons, Inc
70 citations
TL;DR: In this article, it was shown that for a tree T and integer k, 2 ≤ k ≤ n − 1, and the range for μn(T) in terms of n and |V(T)| is established.
Abstract: The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n-distance μn(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μn(G) ≤ μk(G) + μn+1−k(G) where 2 ≤ k ≤ n − 1. The range for the average Steiner n-distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n − 1, it is shown that μn(T) ≤ (n/k)μk(T) and the range for μn(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n-distance of a tree are outlined. © 1996 John Wiley & Sons, Inc.
65 citations
TL;DR: In this paper, it was shown that the number of points with pairwise different sets of neighbors in a graph is O(2r/2), where r is the rank of the adjacency matrix.
Abstract: We show that the number of points with pairwise different sets of neighbors in a graph is O(2r/2), where r is the rank of the adjacency matrix. We also give an example that achieves this bound. © 1996 John Wiley & Sons, Inc.
63 citations
TL;DR: In this paper, a graph G is called well covered if every maximal independent set of G has the same number of vertices, where vertices correspond to two maximal independent sets of G. In this paper, we extend the definition of well covered simplicial, chordal and circular arc graphs.
Abstract: A graph G is called well covered if every two maximal independent sets of G have the same number of vertices. In this paper, we characterize well covered simplicial, chordal and circular arc graphs. © 1996 John Wiley & Sons, Inc.
63 citations
TL;DR: In this article, the size of a graph in G(n, k) with order n and lambda-number k is examined, and it is shown that for any integer n between the maximum and minimum sizes there exists a graph with size n in G (n, K).
Abstract: A labeling of graph G with a condition at distance two is an integer labeling of V(G) such that adjacent vertices have labels that differ by at least two, and vertices distance two apart have labels that differ by at least one. The lambda-number of G, λ(G), is the minimum span over all labelings of G with a condition at distance two. Let G(n, k) denote the set of all graphs with order n and lambda-number k. In this paper, we examine the sizes of graphs in G(n, k). We modify Chvatal's result on non-hamiltonian graphs to obtain a formula for the minimum size of a graph in G(n, k), and we use an algorithmic approach to obtain a formula for the maximum size. Finally, we show that for any integer j between the maximum and minimum sizes there exists a graph with size j in G(n, k). © 1996 John Wiley & Sons, Inc.
57 citations
TL;DR: In this article, the maximum number of two edge colorings of a graph on n vertices that admit no monochromatic Kk was shown to be 2bn /4c for all n ≥ 6.
Abstract: Let F (n, k) denote the maximum number of two edge colorings of a graph on n vertices that admit no monochromatic Kk, (a complete graph on k vertices). The following results are proved: F (n, 3) = 2bn /4c for all n ≥ 6. F (n, k) = 2 k−2 2k−2+o(1))n 2 . In particular, the first result solves a conjecture of Erdos and Rothschild.
54 citations
TL;DR: The problem of determining the domination number of a graph is a well known NP-hard problem, even when restricted to planar graphs as discussed by the authors, and it has been shown that planar graph with diameter two and three can be determined in polynomial time.
Abstract: The problem of determining the domination number of a graph is a well known NP-hard problem, even when restricted to planar graphs. By adding a further restriction on the diameter of the graph, we prove that planar graphs with diameter two and three have bounded domination numbers. This implies that the domination number of such a graph can be determined in polynomial time. We also give examples of planar graphs of diameter four, and nonplanar graphs of diameter two, having arbitrarily large domination numbers. © 1996 John Wiley & Sons, Inc.
TL;DR: Bollobas and Sauer as mentioned in this paper showed that for any integer n ≥ 2, there are graphs G of arbitrary large girth with star-chromatic number χ*(G) = k/d.
Abstract: Suppose G and H are graphs. We say G is H-colorable if there is a homomorphism (edge-preserving vertex mapping) from G to H. We say a graph G is uniquely H-colorable if there is an onto homomorphism c from G to H, and any other homomorphism from G to H is the composition σ ??? c of c with an automorphism σ of H. In case H is the complete graph Kn, the notion of uniquely H-colorable coincides with that of uniquely n-colorable. It was proved by B. Bollobas and N. Sauer that for any integer n there are graphs of arbitrary large girth which are uniquely n-colorable. We generalize this result and prove that for any graph H which is a core (i.e., H admits no homomorphisms to any of its proper subgraphs), and for any integer g, there is a graph which is uniquely H-colorable and has girth at least g. We then use this generalization to answer a question concerning the star-chromatic number of a graph. A graph G is said to be (k,d)-colorable if there is a coloring c of the vertices of G with k colors {0,1,…,k − 1} such that d ≤ |c(x)−c(y)| ≤ k − d for every edge (x,y) of G. The star-chromatic number of G is the infimum of the ratio k/d such that G is (k,d)-colorable. We shall show that for any rational k/d ≥ 2, there are graphs G of arbitrary large girth with star-chromatic number χ*(G) = k/d. In particular, for any integer n, there are graphs G of arbitrary large girth with χ*(G) = χ(G)=n. © 1996 John Wiley & Sons, Inc.
TL;DR: In this paper, the authors give a polynomial time algorithm for approximating the rectilinear crossing number of a graph on a book, where the vertices of the graph are placed on the spine of a k-page book and edges are drawn on pages such that each edge is contained by one page.
Abstract: Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log2n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m2/k2) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m3/n2k2), provided that m = Ψ(n2). © 1996 John Wiley & Sons, Inc.
TL;DR: In this paper, it was shown that any tournament has a node whose outdegree is at least doubled in its square, where the directed distance in the tournament is at most two.
Abstract: Let the square of a tournament be the digraph on the same nodes with arcs where the directed distance in the tournament is at most two. This paper verifies Dean's conjecture: any tournament has a node whose outdegree is at least doubled in its square. © 1996 John Wiley & Sons, Inc.
TL;DR: In this article, it was shown that any two bipartite quadrangulations of any closed surface are transformed into each other by two kinds of transformations, called the diagonal slide and the diagonal rotation, up to homeomorphism, if they have the same and sufficiently large number of vertices.
Abstract: In this paper, it will be shown that any two bipartite quadrangulations of any closed surface are transformed into each other by two kinds of transformations, called the diagonal slide and the diagonal rotation, up to homeomorphism, if they have the same and sufficiently large number of vertices. © 1996 John Wiley & Sons, Inc.
TL;DR: A structural theorem on plane graphs is proved in the present paper which implies the validity of this conjecture for all D(G) ≥ 7 as discussed by the authors, where D is the maximum degree of a graph.
Abstract: In 1973, Kronk and Mitchem (Discrete Math.(5) 255–260) conjectured that the vertices, edges and faces of each plane graph G may be colored with D(G) + 4 colors, where D(G) is the maximum degree of G, so that any two adjacent or incident elements receive distinct colors. They succeeded in verifying this for D(G) = 3. A structural theorem on plane graphs is proved in the present paper which implies the validity of this conjecture for all D(G) ≥ 7. © 1996 John Wiley & Sons, Inc.
TL;DR: In this paper, it was shown that if G and H are arbitrary fixed graphs and n is sufficiently large, then, where k = χ(G) and m = |V(H)|, the largest component has m edges.
Abstract: It is shown that if G and H are arbitrary fixed graphs and n is sufficiently large, then , where k = χ(G) and m = |V(H)|. Also, we prove that for any forest ??? whose largest component has m edges. Thus , where . We conjecture that . © 1996 John Wiley & Sons, Inc.
TL;DR: In this paper, a new type of sufficient condition for a digraph to be Hamiltonian was proposed, which combines local structure of the digraph with conditions on the degrees of non-adjacent vertices.
Abstract: We describe a new type of sufficient condition for a digraph to be Hamiltonian. Conditions of this type combine local structure of the digraph with conditions on the degrees of non-adjacent vertices. The main difference from earlier conditions is that we do not require a degree condition on all pairs of non-adjacent vertices. Our results generalize the classical conditions by Ghouila-Houri and Woodall.
TL;DR: In this article, it was shown that for each k ≥ 4 there exists a connected k-domination critical graph with independent domination number exceeding k, thus disproving a conjecture of Sumner and Blitch (J. Combinatorial Theory B34 (1983), 65-76) in all cases except k = 3.
Abstract: We show that for each k ≥ 4 there exists a connected k-domination critical graph with independent domination number exceeding k, thus disproving a conjecture of Sumner and Blitch (J. Combinatorial Theory B34 (1983), 65–76) in all cases except k = 3. © 1996 John Wiley & Sons, Inc.
TL;DR: The crossing number of K3,n in a surface with Euler genus ϵ is ⌊n/(2 ϵ + 2)⌋ (n − (ϵ + 1) {1 + ⌉ n/(2ϵ+ 2) ⌋}) as discussed by the authors.
Abstract: In this article, we show that the crossing number of K3,n in a surface with Euler genus ϵ is ⌊n/(2ϵ + 2)⌋ (n − (ϵ + 1) {1 + ⌊n/(2ϵ + 2)⌋}). This generalizes a result of Guy and Jenkyns, who obtained this result for the torus. © 1996 John Wiley & Sons, Inc.
TL;DR: A sufficient condition for a 2-connected planar graph G to be Hamiltonian is given in this article, and a linear algorithm to find a Hamilton cycle can be extracted from the proof.
Abstract: Let G be a 2-connected plane graph with outer cycle XG such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of XG. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.
TL;DR: In this article, it was shown that almost every triangle-free graph in Forbn,m (K3) has at least chromatic number 3, provided that c,n 0 are appropriate constants.
Abstract: An important result of Erdos. Kleitman, and Rothsorld says that almost every triangle-free graph on n vertices has chromatic number 2. In this paper we study the asymptotic structure of graphs in Forbn.m (K3), i.e., in the class of triangle-free graphs on n verticles having m = m(n) edges. In particular we prove that an analogue to the Erdos-Kleitman-Rothschild result is true, whenever m ≥ cn7/4 log n for some constant c > 0. On the other hand, it is shown that almost every graph in Forbn,m (K3) has at least chromatic number 3, provided that c,n 0 are appropriate constants. © 1996 John Wiley & Sons, Inc.
TL;DR: In this article, the authors show that odd cycles and their complements are star-superperfect and prove a theorem about the circular chromatic number of lexicographic product of graphs.
Abstract: Suppose that G is a finite simple graph and w is a weight function which assigns to each vertex of G a nonnegative real number. Let C be a circle of length t. A t-circular coloring of (G, w) is a mapping Δ of the vertices of G to arcs of C such that Δ(x)∩Δ(y) = 0 if (x, y) ∈ E(G) and Δ(x) has length w(x). The circular-chromatic number of (G, w) is the least t for which there is a t-circular coloring of (G, w). This paper discusses basic properties of circular chromatic number of a weighted graph and relations between this parameter and other graph parameters. We are particularly interested in graphs G for which the circular-chromatic number of (G, w) is equal to the fractional clique weight of (G, w) for arbitrary weight function w. We call such graphs star-superperfect. We prove that odd cycles and their complements are star-superperfect. We then prove a theorem about the circular chromatic number of lexicographic product of graphs which provides a tool of constructing new star-superperfect graphs from old ones. © 1996 John Wiley & Sons, Inc.