Showing papers on "Path graph published in 1979"
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.
1,312 citations
TL;DR: The proof is conerned with those subgraphs H of a graph G for which there is an edge-preserving map f of the set of vertices of G onto the set.
Abstract: Let G be an undirected graph without multiple edges and with a loop at every vertex—the set of edges of G corresponds to a reflexive and symmetric binary relation on its set of vertices. Then every edge-preserving map of the set of vertices of G to itself fixes an edge [{f(a), f(b)} = {a, b} for some edge (a, b) of G] if and only if (i) G is connected, (ii) G contains no cycles, and (iii) G contains no infinte paths. The proof is conerned with those subgraphs H of a graph G for which there is an edge-preserving map f of the set of vertices of G onto the set of vertices of H and satisfying f(a) = a for each vertex a of H.
92 citations
TL;DR: It is shown that the interval number of a graph on n vertices is at most [14(n+1), and this bound is best possible, and the complete bipartite graph K is attained.
50 citations
TL;DR: A (0,@l)-graph (@l>=2) is a connected graph in which any two vertices have @l common neighbours or none at all, and such a graph is regular when the diameter is at least four.
34 citations
01 Jan 1979
TL;DR: In this article, the authors present sufficient conditions for a set of real numbers to be the spectrum of a nonnegative matrix, in fact, of a symmetric one, which is a difficult problem.
Abstract: This chapter discusses the symmetric nonnegative matrices. The problem of obtaining sufficient conditions for a set of complex numbers to be the spectrum of a nonnegative matrix appears to be a difficult one. The chapter presents sufficient conditions for a set of real numbers to be the spectrum of a nonnegative matrix, in fact, of a symmetric one. A cycle is a chain from a vertex to itself, such that no edge appears twice in the sequence. If V denotes the set of vertices of G and if A is a subset of V , then the graph with A as its vertex set and with all the edges in G that have their endpoints in A is called a subgraph of G . A connected graph is a graph that contains a chain from x to y for each pair x and y of distinct vertices. The relation, x = y or there exists a chain in G from x to y , is an equivalence relation on V . The classes of this equivalence relation partition V into connected subgraphs called the connected components of G . A tree is a connected graph without cycles.
30 citations
TL;DR: It is shown that there exists such a graph with m edges and n vertices and two distinguished vertices such that the number of distinct paths from s to t is maximized, and its structure is investigated.
26 citations
TL;DR: A method is described of constructing a class of self-complementary graphs, that includes a self- complementary graph, containing no K5, with 41 vertices and aSelf-complementation graph,containing no K7, with 113 vertices, that gives the improved Ramsey number lower bound r(7, 7) ≥ 114.
Abstract: A method is described of constructing a class of self-complementary graphs, that includes a self-complementary graph, containing no K5, with 41 vertices and a self-complementary graph, containing no K7, with 113 vertices. The latter construction gives the improved Ramsey number lower bound r(7, 7) ≥ 114.
22 citations
TL;DR: The principal result of this paper provides a nearly complete answer to the following question: for which integers k, m, n is it true that whenever the authors have a graph G whose set of vertices is an ordered set of order type ωn, then this graph contains a complete subgraph with k vertices.
8 citations
8 citations
TL;DR: In this article, the authors give simple proofs of the existence of a matched bridge in a graph with a unique 1-factor and condition that it has at least n totally covered vertices, if n ⩾ 2.
3 citations