Minimum crossings in join of graphs with paths and cycles

Minimum Crossings in Join of Graphs with Paths and Cycles

The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. Only few results concerning crossing numbers of graphs obtained as join product of two graphs are known. There was collected the exact values of crossing numbers for join of all graphs of at most four vertices and of several graphs of order five with paths and cycles. We extend these r...

Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p ≥ ω/k for a function ω = ω(k) that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least (1− o(1))k, and (ii) if p ≥ (1 + o(1)) ln k/k, then Gp asymptotica...

Alternating Paths and Cycles of Minimum Length

Finding simple paths and cycles in graphs

We describe a novel method, the method of random colorings, for finding simple paths and cycles of a specified length k in a graph G = (V,E). Using small bias probability spaces and families of perfect hash functions we can derandomize the randomized algorithms obtained using this method and obtain deterministic algorithms that establish the following new results: • For every fixed k, deciding ...

Finding Paths and Cycles in Graphs

A polynomial time algorithm which detects all paths and cycles of all lengths in form of vertex pairs (start, finish). Introduction Problem of path/cycle existence and finding is a big topic in Discrete Mathematics and Computer Science [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, and others]. This article is not a review but just another design. The design’s idea may be traced back ...