74,608 research outputs found
A bound on the number of edges in graphs without an even cycle
We show that, for each fixed , an -vertex graph not containing a cycle
of length has at most edges.Comment: 16 pages, v2 appeared in Comb. Probab. Comp., v3 fixes an error in v2
and explains why the method in the paper cannot improve the power of k
further, v4 fixes the proof of Theorem 12 introduced in v
MaxCut in graphs with sparse neighborhoods
Let be a graph with edges and let denote the size of
a largest cut of . The difference is called the surplus
of . A fundamental problem in MaxCut is to determine
for without specific structure, and the degree sequence
of plays a key role in getting the lower bound of
. A classical example, given by Shearer, is that
for triangle-free graphs ,
implying that . It was extended to graphs with
sparse neighborhoods by Alon, Krivelevich and Sudakov. In this paper, we
establish a novel and stronger result for a more general family of graphs with
sparse neighborhoods.
Our result can derive many well-known bounds on in -free
graphs for different , such as the triangle, the even cycle, the graphs
having a vertex whose removal makes the graph acyclic, or the complete
bipartite graph with . It can also deduce many new
(tight) bounds on in -free graphs when is any graph
having a vertex whose removal results in a bipartite graph with relatively
small Tur\'{a}n number, especially the even wheel. This contributes to a
conjecture raised by Alon, Krivelevich and Sudakov. Moreover, we give a new
family of graphs such that for
some constant in -free graphs , giving an evidence to a
conjecture suggested by Alon, Bollob\'as, Krivelevich and Sudakov
An exploration of two infinite families of snarks
Thesis (M.S.) University of Alaska Fairbanks, 2019In this paper, we generalize a single example of a snark that admits a drawing with even rotational symmetry into two infinite families using a voltage graph construction techniques derived from cyclic Pseudo-Loupekine snarks. We expose an enforced chirality in coloring the underlying 5-pole that generated the known example, and use this fact to show that the infinite families are in fact snarks. We explore the construction of these families in terms of the blowup construction. We show that a graph in either family with rotational symmetry of order m has automorphism group of order m2m⁺¹. The oddness of graphs in both families is determined exactly, and shown to increase linearly with the order of rotational symmetry.Chapter 1: Introduction -- 1.1 General Graph Theory -- Chapter 2: Introduction to Snarks -- 2.1 History -- 2.2 Motivation -- 2.3 Loupekine Snarks and k-poles -- 2.4 Conditions on Triviality -- Chapter 3: The Construction of Two Families of Snarks -- 3.1 Voltage Graphs and Lifts -- 3.2 The Family of Snarks, Fm -- 3.3 A Second Family of Snarks, Rm -- Chapter 4: Results -- 4.1 Proof that the graphs Fm and Rm are Snarks -- 4.2 Interpreting Fm and Rm as Blowup Graphs -- 4.3 Automorphism Group -- 4.4 Oddness -- Chapter 5: Conclusions and Open Questions -- References
Generalized Tur\'an problems for even cycles
Given a graph and a set of graphs , let
denote the maximum possible number of copies of in an -free
graph on vertices. We investigate the function , when
and members of are cycles. Let denote the cycle of
length and let . Some of our main
results are the following.
(i) We show that for any .
Moreover, we determine it asymptotically in the following cases: We show that
and that the maximum
possible number of 's in a -free bipartite graph is .
(ii) Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds,
then for any we have . We prove that forbidding any other even cycle
decreases the number of 's significantly: For any , we have
More generally,
we show that for any and such that , we have
(iii) We prove provided a
strong version of Erd\H{o}s's Girth Conjecture holds (which is known to be true
when ). Moreover, forbidding one more cycle decreases the number
of 's significantly: More precisely, we have and for .
(iv) We also study the maximum number of paths of given length in a
-free graph, and prove asymptotically sharp bounds in some cases.Comment: 37 Pages; Substantially revised, contains several new results.
Mistakes corrected based on the suggestions of a refere
- …