Jozef Miškuf

Total edge irregularity strength of complete graphs and complete bipartite graphs

A total edge irregular k-labelling @n of a graph G is a labelling of the vertices and edges of G with labels from the set {1,...,k} in such a way that for any two different edges e and f their weights @f(f) and @f(e) are distinct. Here, the weight of an edge g=uv is @f(g)=@n(g)+@n(u)+@n(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G. We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs.

Fullerene graphs have exponentially many perfect matchings

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.

Total Edge Irregularity Strength of Complete Graphs and Complete Bipartite Graphs

Abstract A total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set { 1 , … , k } in such a way that for any two different edges e and f their weights φ ( f ) and φ ( e ) are distinct where the weight of an edge g = u v is φ ( g ) = ν ( e ) + ν ( u ) + ν ( v ) , i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G. We show the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs.

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k@?N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree @D where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most @D+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most @D+1. We also present examples where these bounds are attained. Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most @D(G)-@D(G).

Backbone Colorings and Generalized Mycielski Graphs

For a graph

Edge-Injective and Edge-Surjective Vertex Labellings

G

Looseness of Plane Graphs

and its spanning tree

On a conjecture about edge irregular total labelings

T

Edge irregular total labellings for graphs of linear size

the backbone chromatic number,

Note: List coloring of Cartesian products of graphs

\mathrm{BBC}(G,T)