The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph L(G). The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in L(G). Attributed to Gallai (1967), the study ...

The Gallai graph and the anti-Gallai graph of a graph G have the edges of G as their vertices. Two edges of G are adjacent in the Gallai graph of G if they are incident but do not span a triangle in G; they are adjacent in the anti-Gallai graph of G if they span a triangle in G. In this paper we show: The Four Color Theorem can be equivalently stated in terms of anti-Gallai graphs; the problems...

The paper deals with graph operators—the Gallai graphs and the anti-Gallai graphs. We prove the existence of a finite family of forbidden subgraphs for the Gallai graphs and the anti-Gallai graphs to be H-free for any finite graph H. The case of complement reducible graphs—cographs is discussed in detail. Some relations between the chromatic number, the radius and the diameter of a graph and it...

The Gallai and the anti-Gallai graphs of a graph G are complementary pairs of spanning subgraphs of the line graph of G. In this paper we find some structural relations between these graph classes by finding a partition of the edge set of the line graph of a graph G into the edge sets of the Gallai and anti-Gallai graphs of G. Based on this, an optimal algorithm to find the root graph of a line...

This chapter deals with two graph classes the Gallai graphs and the anti-Gallai graphs. \Ve construct infinitely many pairs of graphs G and H such that f( G) = r( H). The existence of a finite family offorbidden subgraphs for the Gallai graphs and the ant.i-Gallai graphs to be H-frce, for any fillit.e graph H is proved and the forbidden subgraph characterizations of G for which the Gallai graph...

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi) graph is a simple, complete, edge-colored (multi) graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decompos...

In 1959 Gallai [5] showed that the vertex independence number and the vertex covering number of a graph G = (V,E) sum to |V |. Over the last twenty years, many results similar to Gallai’s Theorem have been observed [3]. These theorems are referred to as “Gallai Theorems” and usually have the form: α+ β = n. Slater [17] described several graph subset parameters using linear programs (LP) and int...

In 1966 Gallai asked whether all longest paths in a connected graph intersect in a common vertex. This is not true in general and various counterexamples have been found. However, many well-known classes of graphs have the socalled Gallai property, e.g. trees and outerplanar graphs. Recently, de Rezende et al. proved that 2-trees, which are maximal series-parallel graphs, also have this propert...

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai-Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of ...