14 research outputs found
Some results on concatenating bipartite graphs
We consider two functions and , defined as follows. Let and let be disjoint nonempty subsets of a graph , where every
vertex in has at least neighbors in , and every vertex in has
at least neighbors in . We denote by the maximum such
that, in all such graphs , there is a vertex that is joined to at
least vertices in by two-edge paths. If in addition we require that
every vertex in has at least neighbors in , and every vertex in
has at least neighbors in , we denote by the maximum
such that, in all such graphs , there is a vertex that is
joined to at least vertices in by two-edge paths. In their recent
paper, M. Chudnovsky, P. Hompe, A. Scott, P. Seymour, and S. Spirkl introduced
these functions, proved some general results about them, and analyzed when they
are greater than or equal to and . Here, we extend their
results by analyzing when they are greater than or equal to and
.Comment: arXiv admin note: text overlap with arXiv:1902.10878 by other author
Proof of the Caccetta-Haggkvist conjecture for digraphs with small independence number
For a digraph and , let be the number of
out-neighbors of in . The Caccetta-H\"{a}ggkvist conjecture states that
for all , if is a digraph with such that for all , then G contains a directed cycle of length at
most . In [2], N. Lichiardopol proved that this conjecture is true for
digraphs with independence number equal to two. In this paper, we generalize
that result, proving that the conjecture is true for digraphs with independence
number at most
Aharoni's rainbow cycle conjecture holds up to an additive constant
In 2017, Aharoni proposed the following generalization of the
Caccetta-H\"{a}ggkvist conjecture: if is a simple -vertex edge-colored
graph with color classes of size at least , then contains a rainbow
cycle of length at most .
In this paper, we prove that, for fixed , Aharoni's conjecture holds up to
an additive constant. Specifically, we show that for each fixed ,
there exists a constant such that if is a simple -vertex
edge-colored graph with color classes of size at least , then
contains a rainbow cycle of length at most .Comment: 10 pages, 0 figures. Upgraded the main result from the previous
version so that it now holds up to an additive constan
Further approximations for Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture
For a digraph and , let be the number of
out-neighbors of in . The Caccetta-H\"{a}ggkvist conjecture states that
for all , if is a digraph with such that for all , then contains a directed cycle of length at
most . Aharoni proposed a generalization of this conjecture,
that a simple edge-colored graph on vertices with color classes, each
of size , has a rainbow cycle of length at most . With
Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each
color class has size . In this paper, we present a proof of
the conjecture if each color class has size , which improved the
previous result and is only a constant factor away from Aharoni's conjecture.
We also consider what happens when the condition on the number of colors is
relaxed
Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture
For a digraph and , let be the number of
out-neighbors of in . The Caccetta-H\"{a}ggkvist conjecture states that
for all , if is a digraph with such that for all , then contains a directed cycle of length at
most . Aharoni proposed a generalization of this conjecture,
that a simple edge-colored graph on vertices with color classes, each
of size at least , has a rainbow cycle of length at most . Let us call \emph{triangular} if every simple
edge-colored graph on vertices with at least color classes, each
with at least edges, has a rainbow triangle. Aharoni, Holzman, and
DeVos showed the following: is triangular; is triangular.
In this paper, we improve those bounds, showing the following:
is triangular; is triangular. Our methods give results for
infinitely many pairs , including ; we show that
is triangular.Comment: Accepted manuscript; see DOI for journal versio
Cycles and coloring in graphs and digraphs
We show results in areas related to extremal problems in directed graphs. The first concerns a rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture, made by Aharoni. The Caccetta-H\"{a}ggkvist conjecture states that if is a simple digraph on vertices with minimum out-degree at least , then there exists a directed cycle in of length at most . Aharoni proposed a generalization of this well-known conjecture, namely that if is a simple edge-colored graph (not necessarily properly colored) on vertices with color classes each of size at least , then there exists a rainbow cycle in of length at most .
In this thesis, we first prove that if is an edge-colored graph on vertices with color classes each of size at least , then has a rainbow cycle of length at most . Then, we develop more techniques to prove the stronger result that if there are color classes, each of size at least , then there is a rainbow cycle of length at most . Finally, we improve upon existing bounds for the triangle case, showing that if there are color classes of size at least , then there exists a rainbow triangle, and also if there are color classes of size at least , then there is a rainbow triangle.
Let denote the \emph{chromatic number} of a graph and let denote the \emph{clique number}. Similarly, let \dichi(D) denote the \emph{dichromatic number} of a digraph and let denote the clique number of the underlying undirected graph of . In the second part of this thesis, we consider questions of -boundedness and \dichi-boundedness. In the undirected setting, the question of -boundedness concerns, for a class of graphs, for what functions (if any) is it true that for all graphs . In a similar way, the notion of \dichi-boundedness refers to, given a class of digraphs, for what functions (if any) is it true that \dichi(D) \le f(\omega(D)) for all digraphs . It was a well-known conjecture, sometimes attributed to Esperet, that for all there exists such that in every graph with with and , there exists an induced subgraph of with and . We disprove this conjecture. Then, we examine the class of -chordal digraphs, which are digraphs such that all induced directed cycles have length equal to . We show that for , the class of -chordal digraphs is not \dichi-bounded, generalizing a result of Aboulker, Bousquet, and de Verclos in [1] for . Then we give a hardness result for determining whether a digraph is -chordal, and finally we show a result in the positive direction, namely that the class of digraphs which are -chordal and also do not contain an induced directed path on vertices is \dichi-bounded.
We discuss the work of others stemming from and related to our results in both areas, and outline directions for further work
A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number
We prove that for every , there is a graph with and
such that every induced subgraph of with satisfies .
This disproves a well-known conjecture. Our construction is a digraph with
bounded clique number, large dichromatic number, and no induced directed cycles
of odd length at least 5.Comment: Moving one of the results to a different paper, where it fits bette
Digraphs with All Induced Directed Cycles of the Same Length are not → χ -Bounded
For t > 2, let us call a digraph D t-chordal if all induced directed cycles in D have length equal to t. In an earlier paper, we asked for which t it is true that t-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for t = 3, that is, they gave a construction of 3-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each t > 3 a construction of t-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number, thus answering our question in the negative. On the other hand, we show that a more restricted class, digraphs with no induced directed cycle of length less than t, and no induced directed t-vertex path, have bounded dichromatic number if their clique number is bounded. We also show the following complexity result: for fixed t > 2, the problem of determining whether a digraph is t-chordal is coNP-complete.This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-03912