Binding number and path-factor critical deleted graphs

Crossing-number critical graphs have bounded path-width

The crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G − e) < cr(G) for all edges e of G. We prove that crossing-critical graphs have “bounded path-width” (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined i...

Independent Set Neighborhood Union And Fractional Critical Deleted Graphs∗

A graph G is called a fractional (k, n′,m)-critical deleted graph if any n′ vertices are removed from G the resulting graph is a fractional (k,m)-deleted graph. In this paper, we determine that for integers k ≥ 1, i ≥ 2, n′,m ≥ 0, n > 4ki+ n′ + 4m− 4, and δ(G) ≥ k(i− 1) + n′ + 2m, if |NG(x1) ∪NG(x2) ∪ · · · ∪NG(xi)| ≥ n+ n′ 2 for any independent subset {x1, x2, . . . , xi} of V (G), then G is a...

Crossing-Critical Graphs and Path-Width

Path covering number and L(2,1)-labeling number of graphs

A path covering of a graph G is a set of vertex disjoint paths of G containing all the vertices of G. The path covering number of G, denoted by P (G), is the minimum number of paths in a path covering of G. An k-L(2, 1)-labeling of a graph G is a mapping f from V (G) to the set {0, 1, . . . , k} such that |f(u) − f(v)| ≥ 2 if dG(u, v) = 1 and |f(u) − f(v)| ≥ 1 if dG(u, v) = 2. The L(2, 1)-label...

On the path separation number of graphs

A path separator of a graph G is a set of paths P = {P1, . . . , Pt} such that for every pair of edges e, f ∈ E(G), there exist paths Pe, Pf ∈ P such that e ∈ E(Pe), f 6∈ E(Pe), e 6∈ E(Pf ) and f ∈ E(Pf ). The path separation number of G, denoted psn(G), is the smallest number of paths in a path separator. We shall estimate the path separation number of several graph families, including complet...