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On Maximum Weight of a Bipartite Graph of Given Order and Size

Mirko Horňák; Stanislav Jendrol’; Ingo Schiermeyer

Discussiones Mathematicae Graph Theory (2013)

  • Volume: 33, Issue: 1, page 147-165
  • ISSN: 2083-5892

Abstract

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The weight of an edge xy of a graph is defined to be the sum of degrees of the vertices x and y. The weight of a graph G is the minimum of weights of edges of G. More than twenty years ago Erd˝os was interested in finding the maximum weight of a graph with n vertices and m edges. This paper presents a complete solution of a modification of the above problem in which a graph is required to be bipartite. It is shown that there is a function w*(n,m) such that the optimum weight is either w*(n,m) or w*(n,m) + 1.

How to cite

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Mirko Horňák, Stanislav Jendrol’, and Ingo Schiermeyer. "On Maximum Weight of a Bipartite Graph of Given Order and Size." Discussiones Mathematicae Graph Theory 33.1 (2013): 147-165. <http://eudml.org/doc/267989>.

@article{MirkoHorňák2013,
abstract = {The weight of an edge xy of a graph is defined to be the sum of degrees of the vertices x and y. The weight of a graph G is the minimum of weights of edges of G. More than twenty years ago Erd˝os was interested in finding the maximum weight of a graph with n vertices and m edges. This paper presents a complete solution of a modification of the above problem in which a graph is required to be bipartite. It is shown that there is a function w*(n,m) such that the optimum weight is either w*(n,m) or w*(n,m) + 1.},
author = {Mirko Horňák, Stanislav Jendrol’, Ingo Schiermeyer},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {weight of an edge; weight of a graph; bipartite graph.; bipartite graph},
language = {eng},
number = {1},
pages = {147-165},
title = {On Maximum Weight of a Bipartite Graph of Given Order and Size},
url = {http://eudml.org/doc/267989},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Mirko Horňák
AU - Stanislav Jendrol’
AU - Ingo Schiermeyer
TI - On Maximum Weight of a Bipartite Graph of Given Order and Size
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 1
SP - 147
EP - 165
AB - The weight of an edge xy of a graph is defined to be the sum of degrees of the vertices x and y. The weight of a graph G is the minimum of weights of edges of G. More than twenty years ago Erd˝os was interested in finding the maximum weight of a graph with n vertices and m edges. This paper presents a complete solution of a modification of the above problem in which a graph is required to be bipartite. It is shown that there is a function w*(n,m) such that the optimum weight is either w*(n,m) or w*(n,m) + 1.
LA - eng
KW - weight of an edge; weight of a graph; bipartite graph.; bipartite graph
UR - http://eudml.org/doc/267989
ER -

References

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  2. [2] H. Enomoto and K. Ota, Connected subgraphs with small degree sum in 3-connected planar graphs, J. Graph Theory 30 (1999) 191-203. doi:10.1002/(SICI)1097-0118(199903)30:3h191::AID-JGT4i3.0.CO;2-X[Crossref] Zbl0916.05020
  3. [3] I. Fabrici and S. Jendrol’, Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs Combin. 13 (1997) 245-250. Zbl0891.05025
  4. [4] B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973) 390-408. doi:10.1007/BF02764716[Crossref] Zbl0265.05103
  5. [5] J. Ivančo, The weight of a graph, in: Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity, J. Neˇsetˇril and M. Fiedler (Ed(s)), (North- Holland, Amsterdam, 1992) 113-116. Zbl0773.05066
  6. [6] J. Ivančo and S. Jendrol’, On extremal problems concerning weights of edges of graphs, in: Sets, Graphs and Numbers, G. Hal´asz, L. Lov´asz, D. Mikl´os and T. Sz˝onyi (Ed(s)), (North-Holland, Amsterdam, 1992) 399-410. Zbl0790.05045
  7. [7] E. Jucovič, Strengthening of a theorem about 3-polytopes, Geom. Dedicata 13 (1974) 233-237. doi:10.1007/BF00183214 Zbl0297.52006
  8. [8] S. Jendrol’ and I. Schiermeyer, On a max-min problem concerning weights of edges, Combinatorica 21 (2001) 351-359. doi:10.1007/s004930100001[Crossref] Zbl1012.05091
  9. [9] S. Jendrol’, M. Tuhársky and H.-J. Voss, A Kotzig type theorem for large maps on surfaces, Tatra Mt. Math. Publ. 27 (2003) 153-162. Zbl1062.05044
  10. [10] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat.-Fyz. ˇ Casopis. Slovensk. Akad. Vied 5 (1955) 111-113. 
  11. [11] J. Zaks, Extending Kotzig’s theorem, Israel J. Math. 45 (1983) 281-296. doi:10.1007/BF02804013[Crossref] Zbl0524.05031

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