15 research outputs found
Graph operations and a unified method for kinds of Tur\'an-type problems on paths, cycles and matchings
Let be a connected graph and a graph parameter. We say
that is feasible if satisfies the following
properties: (I) , if
for any , where is the graph obtained by applying Kelmans
operation from to ; (II) for any edge
. Let be a path of order , the
set of all cycles of length at least and a matching containing
independent edges. In this paper, we mainly prove the following three
results: (i) Let and let
. Let be a -connected
-vertex -free graph with the maximum
where is feasible. Then, . (ii) Let and let
. Let be a connected -vertex
-free graph with the maximum where is
feasible. Then, (iii) Let be a connected
-vertex -free graph with the maximum where
is feasible. Then, when and
when . Directly derived from these three main results, we obtain a
series of applications in Tur\'an-type problems, generalized Tur\'an-type
problems, powers of graph degrees in extremal graph theory, and problems
related to spectral radius, and signless Laplacian spectral radius in spectral
graph theory.Comment: V
Robust expansion and hamiltonicity
This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ‘robustly expanding’ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in Kühn and Osthus’s proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments.
In Chapters 3 and 4, we prove that every sufficiently large 3-connected -regular graph on vertices with ≥ n/4 contains a Hamilton cycle. This answers a problem of Bollobás and Häggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this.
Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the largest degree in a sufficiently large graph on n vertices is at least a little larger than /3 + for ≤ /3, then contains the square of a Hamilton cycle
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum