Our paper has two goals: i) We propose the combinatorial approach to facilitate the calculation of the number of spanning trees for five new classes of graphs. ii) We use a new powerful operation (subdivision) to get larger graphs from a given graph. In particular, we derive the explicit formulas for the subdivision of ladder, fan, triangular snake, double triangular snake and the total graph o...

In 2010, Barát and Tóth verified that any r-critical graph with at most r + 4 vertices has a subdivision of Kr. Based in this result, the authors conjectured that, for every positive integer c, there exists a bound r(c) such that for any r, where r > r(c), any r-critical graph on r+ c vertices has a subdivision of Kr. In this note, we verify the validity of this conjecture for c = 5, and show c...

This paper is an overview of what the author has learned about the critical group of a graph, including some new results. In particular we discuss the critical group of a graph in relation to that of its line graph when the original graph is regular. We begin by introducing the critical group from various aspects. We then study the subdivision graph and line graph in relation to the critical gr...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. Total domination subdivision number denoted by sdγt is the minimum number of edges that must be subdivided to increase the total domination number. Here we...

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G...

One of the basic results in graph theory is Dirac’s theorem, that every graph of order n 3 and minimum degree n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree n/2 contains a cycleC then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every conne...

For any k ∈ N, the k-subdivision of a graph G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the mth power of the n-subdivision of G has been introduced as a fractional power of G, denoted by G m n . In this regard, we investig...

Let G be a simple graph. For any k ∈ N , the k−power of G is a simple graph G with vertex set V (G) and edge set {xy : dG(x, y) ≤ k} and the k−subdivision of G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. So we can introduce the m−power of the n−subdivision of G, as a fractional power of G, that is denoted by G m n . In other words G m

For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n . In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n ) = ω(G ...

Given any integers s, t ≥ 2, we show there exists some c = c(s, t) > 0 such that any Ks,t-free graph with average degree d contains a subdivision of a clique with at least cd 1 2 s s−1 vertices. In particular, when s = 2 this resolves in a strong sense the conjecture of Mader in 1999 that every C4-free graph has a subdivision of a clique with order linear in the average degree of the original g...