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1. Mai 2013 · Since we now have an upper bound on the maximum degree of an extremal graph, we focus on e lower bound. Recall that E 0 = 1 2 q (q+ 1) 2 − q. eorem 3. Let q be an even number and let G be a C 4 -free graph on q 2 + q vertices, with maximum degree + 1 or less. Then, if e denotes the number of edges of G, we have e E 0 .
- Frank A. Firke, Peter M. Kosek, Evan D. Nash, Jason Williford
- 2013
1. Jan. 2024 · Let G be a connected graph and k ∈ N.The k-edge-fault diameter of G is defined as the largest diameter of the graphs arising from G by deleting k edges. In this paper we show that the k-edge-fault diameter of a (k + 1)-edge connected graph of order n not containing a 4-cycle cannot exceed 5 k 2 + k + 1 n + 9, and we construct graphs that show that for k + 1 a prime power this bound is close ...
14. Dez. 2015 · Extremal C4-free/C5-free planar graphs. We study the topic of "extremal" planar graphs, defining exP(n, H) to be the maximum number of edges possible in a planar graph on n vertices that does not contain a given graph H as a subgraph. In particular,we examine the case when H is a small cycle,obtaining exP(n,C4) ≤ 15 7 (n − 2) for all n ≥ ...
- Chris Dowden
- 2015
25. Nov. 2021 · Motivated by a longstanding conjecture of Thomassen, we study how large the average degree of a graph needs to be to imply that it contains a C4-free subgraph with average degree at least t. Kühn and Osthus showed that an average degree bound which is double exponential in t is sufficient. We give a short proof of this bound, before reducing it to a single exponential. That is, we show that ...
4. Apr. 2012 · If G c is chordal, then for every s ≥2 the minimal free resolution of \ (I_ {G}^ {s}\) is linear. This note focuses on a wider class of edge ideals: the C 4 -free edge ideals. As we will see, such ideals arise naturally when we study edge ideals whose powers have linear minimal free resolutions. We say that I G is C 4 -free if the complement ...
We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C 4.We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O ((n log n) 1/3) and consequently size O (n 4/3 (log n) 1/3), which are sharp up to constants.
