scholarly journals The Erdős-Hajnal Property for Graphs with No Fixed Cycle as a Pivot-Minor

10.37236/9536 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jaehoon Kim ◽  
Sang-il Oum
Keyword(s):  
Vertex Graph ◽  
Disjoint Sets

We prove that for every integer $k$, there exists $\varepsilon>0$ such that for every $n$-vertex graph with no pivot-minors isomorphic to $C_k$, there exist disjoint sets $A$, $B$ such that $|A|,|B|\ge\varepsilon n$, and $A$ is complete or anticomplete to $B$. This proves the analog of the Erdős-Hajnal conjecture for the class of graphs with no pivot-minors isomorphic to $C_k$.

scholarly journals Hereditary quasirandomness without regularity

2017 ◽  
Vol 164 (3) ◽  
pp. 385-399 ◽  
Author(s):  
DAVID CONLON ◽  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Alternative Proof ◽  
Regularity Lemma ◽  
Original Proof ◽  
Vertex Graph

AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

Irredundance and Maximum Degree in Graphs

1997 ◽  
Vol 6 (2) ◽  
pp. 153-157 ◽  
Author(s):  
E. J. COCKAYNE ◽  
C. M. MYNHARDT
Keyword(s):  
Maximum Degree ◽  
Vertex Graph

It is proved that the smallest cardinality among the maximal irredundant sets in an n–vertex graph with maximum degree Δ([ges ]2) is at least 2n/3Δ. This substantially improves a bound by Bollobás and Cockayne [1]. The class of graphs which attain this bound is characterised.

Learning Parametric Time-Vertex Graph Processes from Incomplete Realizations

2021 ◽  
Author(s):  
Eylem Tugce Guneyi ◽  
Abdullah Canbolat ◽  
Elif Vural
Keyword(s):  
Vertex Graph

scholarly journals An Ordered Turán Problem for Bipartite Graphs

10.37236/2471 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Free Graph ◽  
The Family ◽  
Turan Problem ◽  
Vertex Graph ◽  
Turán Number ◽  
Turán Problem ◽  
Natural Way

Let $F$ be a graph.  A graph $G$ is $F$-free if it does not contain $F$ as a subgraph.  The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices.  The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem.  In this paper we introduce an ordered version of the Turán problem for bipartite graphs.  Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way.  A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$.  A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part.  Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles.  We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$.  In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$.  For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

scholarly journals Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

scholarly journals Capturing the Drunk Robber on a Graph

10.37236/3398 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Natasha Komarov ◽  
Peter Winkler
Keyword(s):  
Random Walk ◽  
Undirected Graph ◽  
Simple Random Walk ◽  
Adjacent Vertex ◽  
Expected Time ◽  
Capture Time ◽  
Vertex Graph

We show that the expected time for a smart "cop"' to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on $G$; the cop sees all and moves as she wishes, with the object of "capturing" the robber—that is, occupying the same vertex—in least expected time. We show that the cop succeeds in expected time no more than $n {+} {\rm o}(n)$. Since there are graphs in which capture time is at least $n {-} o(n)$, this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.

scholarly journals Densities of Minor-Closed Graph Families

10.37236/408 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
David Eppstein
Keyword(s):  
Order Type ◽  
Closed Graph ◽  
Simple Graphs ◽  
Minimal Graphs ◽  
Vertex Graph ◽  
Graph Families ◽  
A Minor ◽  
Cluster Points

We define the limiting density of a minor-closed family of simple graphs $\mathcal{F}$ to be the smallest number $k$ such that every $n$-vertex graph in $\mathcal{F}$ has at most $kn(1+o(1))$ edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least $\omega^\omega$. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, $1$ and $3/2$. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios $i/(i+1)$.

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