A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if for every vertex v ∈ V (G), the subgraph induced by the neighborhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3… (More)
The well-known Chvátal–Erd˝ os Theorem states that every graph G of order at least three with α(G) ≤ κ(G) has a hamiltonian cycle, where α(G) and κ(G) are the independence number and the connectivity of G, respectively. Oberly and Sumner [J. Graph Theory 3 (1979), 351–356] have proved that every connected, locally-connected claw-free graph of order at least… (More)
A proper edge coloring of a simple graph G is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices u and v of G, the set of the colors assigned to the edges incident to u differs from the set of the colors assigned to the edges incident to v. The minimum number of colors required for all vdecs of G is denoted by χ ′ s (G)… (More)
Corrádi and Hajnal  showed that any graph of order at least 3k with minimum degree at least 2k contains k vertex-disjoint cycles. This minimum degree condition is sharp, because the complete bipartite graph K 2k−1,n−2k+1 does not contain k vertex-disjoint cycles. About the existence of vertex-disjoint cycles of the same length, Thomassen  conjectured… (More)
The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if C is a hamiltonian cycle of a graph G, then the square of C is called a hamiltonian square of G. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square,… (More)
Let G be a simple graph of order n, and let ∆(G) and χ ′ (G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ ′ (G) = ∆(G) or ∆(G) + 1. Following this result, G is called edge-chromatic critical if χ ′ (G) = ∆(G) + 1 and χ ′ (G − e) = ∆(G) for every e ∈ E(G). In 1968, Vizing conjectured that if G is edge-chromatic… (More)
In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is… (More)
We show that if G is a graph such that every edge is in two triangles, then G contains a spanning tree with no vertex of degree 2 (a homeomorphically irreducible spanning tree).