Phone: ++420 / 597 325 972 Address: Department of Applied Mathematics VŠB - Technical University of Ostrava 17. listopadu 15 708 33 Ostrava-Poruba Czech Republic
ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component i... more ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K 6n,6n . In [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph K n,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of K n,n into 3-regular graphs some more. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph K_{3n/2,3n/2}.
Fronček and Kovářová provided in [2] and [3]spanning trees of order 2n that factorize K 2n for ev... more Fronček and Kovářová provided in [2] and [3]spanning trees of order 2n that factorize K 2n for every n ≥ 2 and for every feasible diameter d, 3 ≤ d ≤ 2n − 1. We extend their work and give a spanning tree on 2n vertices with a maximum degree D</font >{\it \Delta} that factorize K 2n for every n ≥ 2 and for every feasible 2 £</font > D</font
Graph decompositions into isomorphic copies of a given graph are a well-established topic studied... more Graph decompositions into isomorphic copies of a given graph are a well-established topic studied in both graph theory and design theory. Although spanning tree factorizations may seem to be just a special case of this concept, not many general results are known. We investigate necessary and sufficient conditions for a graph factorization into isomorphic spanning trees to exist.
ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component i... more ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K 6n,6n . In [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph K n,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of K n,n into 3-regular graphs some more. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph K_{3n/2,3n/2}.
ABSTRACT A graph G with k vertices is distance magic if the vertices can be labeled with numbers ... more ABSTRACT A graph G with k vertices is distance magic if the vertices can be labeled with numbers 1,2,⋯,k so that the sum of labels of the neighbors of each vertex is equal to the same constant μ 0 . We present a construction of distance magic graphs arising from arbitrary regular graphs based on an application of magic rectangles. We also solve a problem posed by M. K. Shafiq, G. Ali and R. Simanjuntak [AKCE Int. J. Graphs Comb. 6, No. 1, 191–200 (2009; Zbl 1210.05154)].
ABSTRACT Let G=(V,E) be a graph on n vertices. A bijection fV→{1,2,⋯,n} is called a distance magi... more ABSTRACT Let G=(V,E) be a graph on n vertices. A bijection fV→{1,2,⋯,n} is called a distance magic labeling of G if there exists an integer k such that ∑ u∈N(v) f(u)=k for all v∈V, where N(v) is the set of all vertices adjacent to v. The constant k is the magic constant of f and any graph which admits a distance magic labeling is a distance magic graph. In this paper we solve some of the problems posted in a recent survey paper on distance magic graph labelings by Arumugam et al. We classify all orders n for which a 4-regular distance magic graph exists and by this we also show that there exists a distance magic graph with k=2 t for every integer t≥6.
ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component i... more ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K 6n,6n . In [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph K n,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of K n,n into 3-regular graphs some more. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph K_{3n/2,3n/2}.
Fronček and Kovářová provided in [2] and [3]spanning trees of order 2n that factorize K 2n for ev... more Fronček and Kovářová provided in [2] and [3]spanning trees of order 2n that factorize K 2n for every n ≥ 2 and for every feasible diameter d, 3 ≤ d ≤ 2n − 1. We extend their work and give a spanning tree on 2n vertices with a maximum degree D</font >{\it \Delta} that factorize K 2n for every n ≥ 2 and for every feasible 2 £</font > D</font
Graph decompositions into isomorphic copies of a given graph are a well-established topic studied... more Graph decompositions into isomorphic copies of a given graph are a well-established topic studied in both graph theory and design theory. Although spanning tree factorizations may seem to be just a special case of this concept, not many general results are known. We investigate necessary and sufficient conditions for a graph factorization into isomorphic spanning trees to exist.
ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component i... more ABSTRACT R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K 6n,6n . In [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph K n,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of K n,n into 3-regular graphs some more. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph K_{3n/2,3n/2}.
ABSTRACT A graph G with k vertices is distance magic if the vertices can be labeled with numbers ... more ABSTRACT A graph G with k vertices is distance magic if the vertices can be labeled with numbers 1,2,⋯,k so that the sum of labels of the neighbors of each vertex is equal to the same constant μ 0 . We present a construction of distance magic graphs arising from arbitrary regular graphs based on an application of magic rectangles. We also solve a problem posed by M. K. Shafiq, G. Ali and R. Simanjuntak [AKCE Int. J. Graphs Comb. 6, No. 1, 191–200 (2009; Zbl 1210.05154)].
ABSTRACT Let G=(V,E) be a graph on n vertices. A bijection fV→{1,2,⋯,n} is called a distance magi... more ABSTRACT Let G=(V,E) be a graph on n vertices. A bijection fV→{1,2,⋯,n} is called a distance magic labeling of G if there exists an integer k such that ∑ u∈N(v) f(u)=k for all v∈V, where N(v) is the set of all vertices adjacent to v. The constant k is the magic constant of f and any graph which admits a distance magic labeling is a distance magic graph. In this paper we solve some of the problems posted in a recent survey paper on distance magic graph labelings by Arumugam et al. We classify all orders n for which a 4-regular distance magic graph exists and by this we also show that there exists a distance magic graph with k=2 t for every integer t≥6.
Uploads