Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Brousek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure...

We describe the critical group of the line graph of the complete bipartite graph Kp,q . Data is presented that describes the structure of the critical group of some regular line graphs in terms of the critical group of the original graph. This suggests the existence of a large class of regular graphs with such a property.

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede’s relatively di...

Statistical graphs—in particular, line graphs and bar graphs—are efficient means of communication in a wide range of non-expert settings. In communication settings, statistical graphs do not only serve as visualizations of individual data points but also provide visual access to various aspects of the information contained in data. Moreover, specific types of graphs are better means for providi...

A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. Here we give a construction for all quasi-line graphs; it turns out that there are basically two kinds of connected quasi-line graphs, one a generalization of line graphs, and the other ...

Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper shows a vertex ordering characterization of simple-triangle graphs as follows: a graph is a simple-triangle graph if and only if there is a linear ordering of ...

Abstract A topological relationship is a key factor in the spatial data processing. Aiming at problem of similarity measurement when dimension objects changes process map generalization, this paper proposes method based on global concept neighborhood graph. Firstly, all relationships that can be described by nine intersection model are abstracted, generalized, and classified, classified point/p...

Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion of color-line graphs that generalizes the classical concept of line graphs in a natural way. Let H be a (properly) edge-colored graph. The (proper) color-line graph CL(H) of H has edges of H as vertices, and two edges of H are adjacent in CL(H) if they are incident in H or have the same color. We gi...

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness.

An r-dynamic coloring of a graph G is proper such that every vertex in V(G) has neighbors at least $\min\{d(v),r\}$ different color classes. The chromatic number denoted as $\chi_r (G)$, the k coloring. In this paper we obtain central graph, middle total line para-line and sub-division comb $P_n\odot K_1$ by $C(P_n\odot K_1), M(P_n\odot T(P_n\odot L(P_n\odot P(P_n\odot K_1)$ $S(P_n\odot respect...