The super line graph of index r of a graph G has the r-sets of edges as its vertices, with two being adjacent if one has an edge that is adjacent to an edge in the other. In this paper, we continue our investigation of this graph by establishing two results, the first on the independence number of super line graphs of arbitrary index and the second on pancyclicity in the index-2 case.

In this paper, we address the minimum 2-edge connected spanning subgraph problem and the graph-TSP in regular bipartite graphs. For these problems, we present new approximation algorithms, each of which finds a restricted 2-factor close to a Hamilton cycle in the first step. We first prove that every regular bipartite graph of degree at least three has a square-free 2-factor. This immediately l...

Let [n]∗ denote the set of integers {−n−1 2 , . . . , n−1 2 } if n is odd, and {−n 2 , . . . , n 2 } \ {0} if n is even. A super edge-graceful labeling f of a graph G of order p and size q is a bijection f : E(G) → [q]∗, such that the induced vertex labeling f ∗ given by f ∗(u) = ∑ uv E(G) f(uv) is a bijection f ∗ : V (G) → [p]∗. A graph is super edge-graceful if it has a super edge-graceful la...

We construct highly edge-connected $r$-regular graph which do not contain $r-2$ pairwise disjoint perfect matchings. The results partially answer a question stated by Thomassen [Factorizing regular graphs, J. Comb. Theory Ser. B (2019), https://doi.org/10.1016/j.jctb.2019.05.002 (article in press)].

For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity, (P2(G)) of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is superif each minimum edge cut is trivial, and it is optimum...

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

For a connected graph G = (V , E), an edge set S ⊂ E is a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum krestricted edge cut. For U1,U2 ⊂ V (G), denote the set of edges of Gwith one end in U1 and the other in U2 by [U1,U2]. Define ξk(G...

Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Brousek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure...

Let G be a graph. For each vertex v 2V (G), Nv denotes the subgraph induces by the vertices adjacent to v inG. The graphG is locally k edge-connected if for each vertex v 2V (G), Nv is k-edge-connected. In this paper we study the existence of nowhere-zero 3-flows in locally k-edgeconnected graphs. In particular, we show that every 2-edge-connected, locally 3-edge-connected graph admits a nowher...