For any two 2-regular spanning subgraphs G and H of the complete multipartite graph K, necessary and sufficient conditions are found for the existence of a 2-factorization of K in which (1) the first and second 2-factors are isomorphic to G and H respectively, and (2) each other 2-factor is a hamilton cycle, in the case where K has an odd number of vertices.

Distance-hereditary graphs (sensu Howorka) are connected graphs in which all induced paths are isometric. Examples of such graphs are provided by complete multipartite graphs and ptolemaic graphs. Every finite distance-hereditary graph is obtained from K, by iterating the following two operations: adding pendant vertices and splitting vertices. Moreover, distance-hereditary graphs are character...

We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erdős, we prove that, if the minimum degree of G is at least 17|V (G)|/20, the largest triangle-free subgraphs are precisely the largest bipartite subgraphs in G. We investigate in particular the case where G is a complete multipartite graph. We prove that a finite trip...

The complete multipartite graph with d partite sets, each containing m vertices, will be denoted by K(d; m). If all of the partite sets are not of the same cardinality, the graph K(mi, m2, . . . . md) is the complete d-partite graph with m, vertices in each partite sets, 1 <i<d. The special case where mi= 1 for every i, yields the complete graph on d vertices and is denoted by Kd. The join of g...

The domination game on a graph G (introduced by B. Brešar, S. Klavžar, D.F. Rall [1]) consists of two players, Dominator and Staller, who take turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay this process as much as po...

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic number is determined for every complete multipartite graph as...

An edge-coloured graph G is called rainbow-connected if any two vertices are connected by a path whose edges have different colours. This concept of rainbow connection in graphs was recently introduced by Chartrand et al. in [4]. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. An easy ob...

The Wiener index of a connected graph G, denoted by W (G), is defined as 12 ∑ u,v∈V (G) dG(u, v). Similarly, the hyper-Wiener index of a connected graph G, denoted by WW (G), is defined as 1 2W (G) + 1 4 ∑ u,v∈V (G) dG(u, v). The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, ...

• Let G be a graph. For subset X of V ( ), the switching σ is signed graph obtained from by reversing signs all edges between and ) ∖ . A( adjacency matrix An eigenvalue called main if it has an eigenvector sum whose entries not equal to zero. Two equivalent graphs share same spectrum, while they may have different eigenvalues. Akbari et al. (2021) conjectured that let ≠ K 2 , 4 { e } Then ther...

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOMP(H). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism probl...