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.
@article{bwmeta1.element.doi-10_7151_dmgt_1674,
author = {Mirko Hor\v n\'ak and Stanislav Jendrol' and Ingo Schiermeyer},
title = {On Maximum Weight of a Bipartite Graph of Given Order and Size},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {147-165},
zbl = {1291.05085},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1674}
}
Mirko Horňák; Stanislav Jendrol’; Ingo Schiermeyer. On Maximum Weight of a Bipartite Graph of Given Order and Size. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 147-165. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1674/
[1] O.V. Borodin, Computing light edges in planar graphs, in: Topics in Combinatorics and Graph Theory, R. Bodendiek and R. Henn (Ed(s)), (Physica-Verlag, Heidelberg, 1990) 137-144.
[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] | Zbl 0916.05020
[3] I. Fabrici and S. Jendrol’, Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs Combin. 13 (1997) 245-250. | Zbl 0891.05025
[4] B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973) 390-408. doi:10.1007/BF02764716[Crossref] | Zbl 0265.05103
[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. | Zbl 0773.05066
[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. | Zbl 0790.05045
[7] E. Jucovič, Strengthening of a theorem about 3-polytopes, Geom. Dedicata 13 (1974) 233-237. doi:10.1007/BF00183214 | Zbl 0297.52006
[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] | Zbl 1012.05091
[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. | Zbl 1062.05044
[10] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat.-Fyz. ˇ Casopis. Slovensk. Akad. Vied 5 (1955) 111-113.
[11] J. Zaks, Extending Kotzig’s theorem, Israel J. Math. 45 (1983) 281-296. doi:10.1007/BF02804013[Crossref] | Zbl 0524.05031