In this paper, we study the minimum cycle base of the planar graphs obtained from two 2-connected planar graphs by identifying an edge (or a cycle) of one graph with the corresponding edge (or cycle) of another, related with map geometries, i.e., Smarandache 2-dimensional manifolds. Also, we give a formula for calculating the length of minimum cycle base of a planar graph N(d, λ) defined in pap...

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach permits the extension to infinite graphs of standard results about fini...

A digraph is a symmetric cycle if it is symmetric and its underlying graph is a cycle. It is proved that if D is an asymmetric digraph not containing a symmetric cycle, then D remains asymmetric after removing some vertex. It is also showed that each digraph D without a symmetric cycle, whose underlying graph is connected, contains a vertex which is a common fixed point of all automorphisms of D.

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G − V (S). For an integer k ≥ 4, a graph G has the single k-cycle property if every edge of G, which does not lie in a triangle, lies in a cycle C of order at most k such that C has at least |V (C)| 2 edges which do not lie in...

In this paper, we consider the edge searching problem on cycle-disjoint graphs. We first improve the running time of the algorithm to compute the vertex separation and the optimal layout of a unicyclic graph, which is given by Ellis et al. (2004), from O(n log n) to O(n). By a linear-time transformation, we can compute the edge search number of a unicyclic graph in linear time. We also propose ...

Let V(G) and E(G) denote the vertex and edge sets of a graph G respectively. Let Z, = (0, 1, . . . , n l}. Let K,, be the complete graph on n vertices. An m-cycle is a simple graph with m vertices, say uo, . . . , u,__~ in which the only edges are uou,_i and the edges joining ui to Ui+l (for 0 s i 6 m 2). We represent this cycle by (uo, . . . , u,_J or (uo, u,_~, u,_~, . . . , ul) or any cyclic...

Jeager et al introduced a concept of group connectivity as an generalization of nowhere zero flows and its dual concept group coloring, and conjectured that every 5-edge connected graph is Z3-connected. For planar graphs, this is equivalent to that every planar graph with girth at least 5 must have group chromatic number at most 3. In this paper we show that if G is a plane graph with girth at ...

Graph labeling is an important area of research in Graph theory. There are many kinds of graph labeling such as Graceful labeling, Magic labeling, Prime labeling, and other different labeling techniques.In this paper the Prime labeling of certain classes of graphs are discussed.It is of interest to note that H-graph which is a 3 –regular graph satisfy Prime labeling. A Gear graph is a graph obt...

This paper generalises the concept of vertex pancyclic graphs. We define a graph as set-pancyclic if for every set S of vertices there is a cycle of every possible length containing S. We show that if the minimum degree of a graph exceeds half its order then the graph is set-pancyclic. We define a graph as k-ordered-pancyclic if, for every set S of cardinality k and every cyclic ordering of S, ...