Showing papers in "Journal of Graph Theory in 1981"
TL;DR: This paper reports on the properties of the matching polynomial α(G) of a graph G, and presents a number of recursion formulas from which it follows that many families of orthogonal polynomials arise as matches of suitable families of graphs.
Abstract: In this paper we report on the properties of the matching polynomial α(G) of a graph G. We present a number of recursion formulas for α(G), from which it follows that many families of orthogonal polynomials arise as matching polynomials of suitable families of graphs. We consider the relation between the matching and characteristic polynomials of a graph. Finally, we consider results which provide information on the zeros of α(G).
246 citations
TL;DR: The main subjects of this survey paper are Hamitonian cycles, cycles of prescirbed lengths, cycles in tournaments, and partitions, packings, and coverings by cycles.
Abstract: The main subjects of this survey paper are Hamitonian cycles, cycles of prescirbed lengths, cycles in tournaments, and partitions, packings, and coverings by cycles. Several unsolved problems and a bibiligraphy are included.
198 citations
TL;DR: Let I ( S ) denote the set of graphs, each with no valency 2 vertices, which are irreducible for S, and using this notation the authors state Kuratowski's theorem.
Abstract: An embedding of a graph G into a surface S is a realization of G as a subspace of S . A graph G is irreducible for S if G does not embed in S , but any proper subgraph of G does embed in S. Irreducible graphs are the smallest (with respect to containment) graphs which fail to embed on a given surface. Let I ( S ) denote the set of graphs, each with no valency 2 vertices, which are irreducible for S . Using this notation we state Kuratowski’s theorem [ 71:
189 citations
TL;DR: This paper shows that if G is a graph with vertex v then there is a tree T with vertex w such that α(G\v, 1/x)/xα(G,1/x) is the generating function for a certain class of walks in G.
Abstract: The matching polynomial α(G, x) of a graph G is a form of the generating function for the number of sets of k independent edges of G. in this paper we show that if G is a graph with vertex v then there is a tree T with vertex w such that
This result has a number of consequences. Here we use it to prove that α(G\v, 1/x)/xα(G, 1/x) is the generating function for a certain class of walks in G. As an application of these results we then establish some new properties of α(G, x).
123 citations
TL;DR: It is shown that every k-connected graph with no 3-cycle contains an edge whose contraction results in a k- Connected graph and this is used to prove that every (k + 3)-connected graph contains a cycle whose deletion results in an k- connected graph.
Abstract: We show that every k-connected graph with no 3-cycle contains an edge whose contraction results in a k-connected graph and use this to prove that every (k + 3)-connected graph contains a cycle whose deletion results in a k-connected graph. This settles a problem of L. Lovasz.
111 citations
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TL;DR: Three short proofs of Kuratowski's theorem on planarity of graphs are presented and applications, extensions, and some related problems are discussed.
Abstract: We present three short proofs of Kuratowski's theorem on planarity of graphs and discuss applications, extensions, and some related problems.
101 citations
TL;DR: A connected graph G is ptolemaic provided that for each four vertices Ui, 1 ≤ i ≤ 4, of G, the six distances dii = dG (Ui, Ui), i ≠ j satisfy the inequality d12d34 ≤ d13d24 + d14d23.
Abstract: A connected graph G is ptolemaic provided that for each four vertices Ui, 1 ≤ i ≤ 4, of G, the six distances dii = dG (Ui, Ui), i ≠ j satisfy the inequality d12d34 ≤ d13d24 + d14d23 (shown by Ptolemy to hold in Euclidean spaces). Ptolemaic graphs were first investigated by Chartrand and Kay, who showed that weakly geodetic ptolemaic graphs are precisely Husimi trees (in particular, trees are ptolemaic). in the present paper several characterizations of ptolemaic graphs are given. It is shown, for example, that a connected graph G is ptolemaic if and only iffor each nondisjoint cliques P, Q of G, their intersection is a cutset of G which separates P-Q and Q-P. An operation is exhibited which generates all finite ptolemaic graphs from complete graphs.
98 citations
TL;DR: A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.
Abstract: A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.
88 citations
TL;DR: Two characterizations are given here of the possible classes of balanced circles of a signed graph: an elementary one of the balanced portion of an arbitrary subclass of circles, and a strongerOne of the entire balanced circle class.
Abstract: The possible classes of balanced circles of a signed graph are characterized in two ways. A signed graph is a graph with arcs signed f or -; a circle is balanced if the product of its arc signs is +. I give here two characterizations of the possible classes of balanced circles of a signed graph: an elementary one of the balanced portion of an arbitrary subclass of circles, and a stronger one of the entire balanced circle class. The latter characterizes signed graphs among biased graphs (explained in [9]). Terminology. A signed graph C consists of an ordinary graph r (finite or infinite) with node set N and arc set E , and a mapping u: E (+, -}, the sign labeling. Loops and multiple arcs are allowed (but we omit the half arcs and free loops needed in other parts of signed graph theory [S]). A path has a Value obtained by multiplying the signs of its constituent arcs; a circle whose value is + is called balanced. An arc set is called balanced when every circle in it is balanced. The class of circles of r is denotedqr); the class of circles balanced in C. is written B ( C ) . (Signed graphs and balance were first conceived by Harary [3].) See Figure 1 for illustrations of signed graphs. First Characterization. When is a class of circles equal t o g @ ) for some C? A generalization: given a certain class 9 of circles of r, when is a subclass @ equal to the balanced subclass o f 9 in some sign labeling of r? To solve the problem we look first at the binary vector space 9 of all subsets of E(T), whose addition is the symmetric difference A. I f 9 C9, we can speak of independent and spanning subsets of 53 (“spanning” means spanning 9). To say a subset 9 is additive in 53 means that whenever C, C1, ..., C, €9 and C = C1 A * A C,, then C €93 if and only if an even number of C , , . . . , C,. are not in 93. This is equivalent to saying that equals either 9 or the intersection of 9 with a hyperplane (codimension *Research assisted by support from the NSF and SGPNR Journal of Graph Theory, Vol. 5 (1981) 401-406 D 1981 by John Wiley & Sons, Inc. CCCC 0364-9024/81/040401-06$01 .OO 402 JOURNAL OF GRAPH THEORY
84 citations
TL;DR: Several sufficient conditions on the degrees of an oriented graph are obtained for the existence of long paths and cycles and that a regular bipartite tournament is hamiltonian.
Abstract: We obtain several sufficient conditions on the degrees of an oriented graph for the existence of long paths and cycles. As corollaries of our results we deduce that a regular tournament contains an edge-disjoint Hamilton cycle and path, and that a regular bipartite tournament is hamiltonian.
77 citations
TL;DR: It is shown that, for each n, all sufficiently large Paley graphs satisfy Axiom n, which concludes at once that several properties of graphs are not first order, including self-complementarity and regularity.
Abstract: A graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, all sufficiently large Paley graphs satisfy Axiom n. From this we conclude at once that several properties of graphs are not first order, including self-complementarity and regularity.
TL;DR: This work examines the problem of embedding a graph H as the center of a supergraph G, and it is demonstrated that A(H) ≤ 4 for all H, and for 0 ≤ i ≤ 4 the authors characterize the class of trees T with A(T) = i.
Abstract: We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference ∣V(G)∣ - ∣V(H)∣ over graphs G having center isomorphic to H it is demonstrated that A(H) ≤ 4 for all H, and for 0 ≤ i ≤ 4 we characterize the class of trees T with A(T) = i. for n ≥ 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n, with chromatic number X(G) = n + X(H), and whose center is isomorphic to H. Finally, if ∣V(H)∣ ≥ 9 and k ≥ ∣V(H)∣ + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.
TL;DR: A proof of that conjecture and a corollary that helps determine the chromatic index of some graphs with 2s points and maximum degree 2s − 2 are presented.
Abstract: Vizing's Theorem states that any graph G has chromatic index either the maximum degree Δ(G) or Δ(G) + 1. If G has 2s + 1 points and Δ(G) = 2s, a well-known necessary condition for the chromatic index to equal 2s is that G have at most 2s2 lines. Hilton conjectured that this condition is also sufficient. We present a proof of that conjecture and a corollary that helps determine the chromatic index of some graphs with 2s points and maximum degree 2s − 2.
TL;DR: Several (nonspectral) classical theorems about line graphs are extended to generalized line graphs, including the derivation and construction of the 31 minimal nongeneralized line graph, a Krausz-type covering characterization, and Whitney-type theorem on isomorphisms and automorphisms.
Abstract: Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. They were originally motivated by spectral considerations. in this paper several (nonspectral) classical theorems about line graphs are extended to generalized line graphs, including the derivation and construction of the 31 minimal nongeneralized line graphs, a Krausz-type covering characterization, and Whitney-type theorems on isomorphisms and automorphisms.
TL;DR: A general enumerative scheme for nonisomorphic maps (up to orientation-preserving homeomorphisms) is developed, which reduces the enumeration of non isomorphic maps of a given type to that of rooted maps (as defined in [4]) of the same and several related kinds.
Abstract: The maps that we count are the planar maps considered in [4]. Two maps are called isomorphic if there is a homeomorphism of the sphere which preserves its orientation, considered as fixed, and takes one map into the other. We develop a general enumerative scheme for nonisomorphic maps (up to orientation-preserving homeomorphisms). Under some conditions it reduces the enumeration of nonisomorphic maps of a given type to that of rooted maps (as defined in [4]) of the same and several related kinds. This partially fills a rather surprising gap between the well-developed techniques for counting rooted maps (cf. [4,5]) and the very few results so far obtained for counting nonisomorphic ones. This note contains a sketch of the method, which is somewhat similar to that developed in [3], and one of the main results. The scheme is based on Burnside’s lemma and on two well-known properties-one topological and the other combinatorial-of a nontrivial orientation-preserving map automorphism [ 1,2,6]: it may be uniquely represented as a rotation of the sphere around an axis (up to topological equivalence-see [ 11) and as a regular permutation acting on the set of halfedges, called “bits” [6 ] or “darts” [2]. If necessary we label the half-edges and consider labeled maps thus obtained. As a principal tool of reduction we use the topologicaI notion of a quotient map of a map with respect to an orientation-preserving automorphism (see [2] for the general definition and properties of quotient maps). It is worth noting that a quotient map is a usual map or a map having 1 or 2 special vertices (free points [2]) of degree 1 which are distinguished as
TL;DR: This work gives a regular graph of girth 6 and valency 7, which has 90 vertices and shows that this is the unique smallest graph with these properties.
Abstract: With the aid of a computer. we give a regular graph of girth 6 and valency 7, which has 90 vertices and show that this is the unique smallest graph with these properties.
TL;DR: An error is pointed out in the generating procedure and an additional operation is included to correct it by including an additionaloperation in the generated procedure.
Abstract: It has been communicated by P. Manca in this journal that all 4-regular connected planar graphs can be generated from the graph of the octahedron using simple planar graph operations. We point out an error in the generating procedure and correct it by including an additional operation.
TL;DR: Sufficient conditions on the degrees of a graph are given in order that its line graph have a hamiltonian cycle.
Abstract: Sufficient conditions on the degrees of a graph are given in order that its line graph have a hamiltonian cycle.
TL;DR: In this paper, a necessary and sufficient condition for a graph H to be the clique graph of some graph G without multicliqual edges was given, and it was shown that there are infinitely many self-clique graphs having more than one critical generator.
Abstract: An edge which belongs to more than one clique of a given graph is called a multicliqual edge. We find a necessary and sufficient condition for a graph H to be the clique graph of some graph G without multicliqual edges. We also give a characterization of graphs without multicliqual edges that have a unique critical generator. Finally, it is shown that there are infinitely many self-clique graphs having more than one critical generator.
TL;DR: This series characterize all the graphs G such that both G and G have the same number of endpoints, and finds that this number can only be 0 or 1 or 2, and is able to enumerate the self-complementary blocks.
Abstract: In this series, we investigate the conditions under which both a graph G and its complement G possess certain specified properties. We now characterize all the graphs G such that both G and G have the same number of endpoints, and find that this number can only be 0 or 1 or 2. As a consequence, we are able to enumerate the self-complementary blocks.
TL;DR: The orientable genus, γ(G), satisfies the inequalities and it is shown that the resulting three values for γ (G1 ∪{x, y}G2) which are possible can actually be realized by appropriate choices for G1 and G2.
Abstract: A graph G is called the 2-amalgamation of subgraphs G1 and G2 if G = G1 ∪ G2 and G1 ∩ G2 = {x, y}, 2 distinct points. in this case we write G = G1∪{x, y}G2. in this paper we show that the orientable genus, γ(G), satisfies the inequalities γ(G1) + γ(G2) − 1 ≤ γ(G1 ∪{x, y}G2) ≤ γ(G1) + γ(G2) + 1 and that this is the best possible result, i. e., the resulting three values for γ(G1 ∪{x, y}G2) which are possible can actually be realized by appropriate choices for G1 and G2.
TL;DR: This work constructs infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex, and constructs, for each n, graphs containing a subset of vertices of size n which are mutually Pseudosimilar.
Abstract: : Dissimilar vertices whose removal leaves isomorphic subgraphs are called pseudosimilar We construct infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex Potential impact on the Reconstruction Conjecture is considered We also construct, for each n, graphs containing a subset of vertices of size n which are mutually pseudosimilar The analogous problem for mutually pseudosimilar edges is introduced (Author)
TL;DR: It is proved that every connected, locally connected graph is upper embeddable and a lower bound for the maximum genus of the square of aconnected graph is given.
Abstract: In this Note it is proved that every connected, locally connected graph is upper embeddable. Moreover, a lower bound for the maximum genus of the square of a connected graph is given.
TL;DR: This Note gives a method for constructing locally homogeneous graphs from groups and the graphs constructable in this way are exactly the locally homogeneity graphs with a point symmetric universal cover.
Abstract: A graph is called locally homogeneous if the subgraphs induced at any two points are isomorphic. in this Note we give a method for constructing locally homogeneous graphs from groups. the graphs constructable in this way are exactly the locally homogeneous graphs with a point symmetric universal cover. As an example we characterize the graphs that are locally n-cycles.
TL;DR: In this article, a lower bound for the Ramsey number r(k) was established, where r ≥ exp (c(log k)4/3[(log log k)1/3] for some constant c> 0.
Abstract: Let r(k) denote the least integer n-such that for any graph G on n vertices either G or its complement G contains a complete graph Kk on k vertices. in this paper, we prove the following lower bound for the Ramsey number r(k) by explicit construction: r(k) ≥ exp (c(Log k)4/3[(log log k)1/3] for some constant c> 0.
TL;DR: Direct proofs of some planarity criteria are presented and it is shown that some of the criteria for planarity can be modified for other criteria.
Abstract: Direct proofs of some planarity criteria are presented.
TL;DR: The set Fn of all pairs (a, b) of integers such that there is a graph G with n vertices and binding number a/b that has a realizing set of b vertices is characterized.
Abstract: The concept of the binding number of a graph was introduced by Woodall in 1973. in this paper we characterize the set Fn of all pairs (a, b) of integers such that there is a graph G with n vertices and binding number a/b that has a realizing set of b vertices.
TL;DR: It is proved that every locally finite, infinite tree not containing a subdivision of the dyadic tree is uniquely determined, up to isomorphism, from its collection of vertex-deleted subgraphs.
Abstract: We prove a theorem saying, when taken together with previous results of Bondy, Hemminger, and Thomassen, that every locally finite, infinite tree not containing a subdivision of the dyadic tree (i. e., the regular tree of degree 3) is uniquely determined, up to isomorphism, from its collection of vertex-deleted subgraphs. Furthermore, as another partial result concerning the reconstruction of locally finite trees, we show that the same is true for locally finite trees whose set of vertices of degree s is nonempty and finite (for some positive integer s).
TL;DR: A graph G has property A(m, n, k) if for any sequence of m + n distinct points of G, there are at least k other points, each of which is adjacent to the first m points of the sequence but not adjacent to any of the latter n points.
Abstract: A graph G has property A(m, n, k) if for any sequence of m + n distinct points of G, there are at least k other points, each of which is adjacent to the first m points of the sequence but not adjacent to any of the latter n points. the minimum order among all graphs with property A(m, n, k) is denoted a(m, n, k). Bounds are given on the numbers a(m, n, k) and some exact results are indicated.