Journal Article10.1007/BF02024498
On maximal paths and circuits of graphs
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About: This article is published in Acta Mathematica Hungarica. The article was published on 01 Sep 1959.
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Citations
Unavoidable chromatic patterns in 2‐colorings of the complete graph
TL;DR: In this article, the authors consider the problem of finding a bipartite omnitonal graph from a set of colorings of the edges of the complete graph, and show that there exists a function (n,G) such that the following holds true for any color: for any 2-coloring $f: E(K_n) \to \{red, blue \}$ such that there are more than n,G$ edges from each color, and for any pair of nonnegative integers $r$ and $b$ with $r+b
Large cycles in graphs
TL;DR: A conjecture of P. Erdos that every graph of order n and size at least 12(n^2-5n+14) has a cycle of length n-1 is proved and a lower bound for the circumference of a non-separable graph in terms of its vertex degrees is given.
Cycles in weighted graphs
J. A. Bondy,Genghua Fan +1 more
TL;DR: It is proved that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n−1) and this generalizes, to weighted graphs, a classical result of Erdős and Gallai.
A Spectral Bipartite Analogue of the Erd\H{O}S-S\'{O}S Conjecture
geng xianya,Wei Wei,Zhiming Feng +2 more
- 01 Jan 2024
Disjoint perfect matchings in 3‐uniform hypergraphs
Abstract: For a hypergraph H, let δ1(H) denote the minimum vertex degree in H. Kühn, Osthus, and Treglown proved that, for any sufficiently large integer n with n≡0(mod3) , if H is a 3‐uniform hypergraph with order n and δ1(H)>n−12−2n/32 then H has a perfect matching, and this bound on δ1(H) is best possible. In this article, we show that under the same conditions, H contains at least ⌈(2n+3)/9⌉ pairwise disjoint perfect matchings, and this bound is sharp.
References
On the structure of linear graphs
Paul Erdös,A. H. Stone +1 more
TL;DR: The first result in this direction was due to Turân as discussed by the authors, who proved that a graph with kn vertices and Ck, 2n+1 edges always contains a complete graph of order k + 1.