Small Edge Sets Meeting all Triangles of a Graph

Abstract

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for 'triangle-3-colorable' graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K 4 -free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.

Induced cycles in triangle graphs

“…The vertex sticking of u and v means removing u and v and introducing a new vertex w adjacent to the entire N(u) ∪ N(v). 1 The inverses of these operations can also be introduced in a natural way.…”

Packing and Covering Directed Triangles

“…The best general result on Tuza's conjecture is due to Haxell [3], who proved that τ (G) ≤ 2.87ν(G) for every graph G. Other authors have approached the conjecture by proving that τ (G) ≤ 2ν(G) for all graphs in some given family. Tuza [10] showed that his conjecture holds for all planar graphs, and Aparna Lakshmanan, Bujtás, and Tuza [8] showed that it holds for all 4-colorable graphs. The planar result has been generalized to graphs without K 3,3 -subdivisions (Krivelevich [6]), and then to graphs with maximum average degree less than 7 (Puleo [7]).…”

Eulerian and Hamiltonian Properties of Gallai and Anti-Gallai Total Graphs

“…Gallai graphs are also used in polynomial time algorithm to recognize k 1,3 -free perfect graphs by Chvatal and Sbihi [15]. Several properties of Gallai and anti-Gallai graphs are discussed in [1], [2], [3], [4], [5], [6] and [18].…”