Another Proof of the Generalized Tutte--Berge Formula for ff-Bounded Subgraphs

Given a nonnegative integer weight $f(v)$ for each vertex $v$ in a multigraph $G$, an {\it $f$-bounded subgraph} of $G$ is a multigraph $H$ contained in $G$ such that $d_H(v)\le f(v)$ for all $v\in V(G)$. Using Tutte's $f$-Factor Theorem, we give a new proof of the min-max relation for the maximum size of an $f$-bounded subgraph of $G$. When $f(v)=1$ for all $v$, the formula reduces to the classical Tutte--Berge Formula for the maximum size of a matching.Comment: 7 page

Combinatorial Generalizations of Sieve Methods and Characterizing Hamiltonicity via Induced Subgraphs

A sieve method is in effect an application of the inclusion-exclusion counting principle, and the estimation methods to avoid computing the explicit formula. Sieve methods have been used in number theory for over a hundred years. These methods have been modified to make use of the structure of integer-like objects; producing better estimates and providing more use cases. The first part of the thesis aims to analyze and use the analogues of number theoretic sieves in combinatorial contexts. This part consists of my work with Yu-Ru Liu in Chapters 2 and 3. We focus on two sieve methods: the Turán sieve (introduced by Liu and Murty in 2005) and the Selberg sieve (independently generalized by Wilson in 1969 and Chow in 1998 with slightly different formulations). Some comparisons and applications of these sieves are discussed. In particular, we apply the combinatorial Turán sieve to count labelled graphs and we apply the combinatorial Selberg sieve to count subspaces of finite spaces. Finding sufficient conditions for Hamiltonicity in graphs is a classical topic, where the difficulty is bracketed by the NP-hardness of the associated decision problem. The second part of the thesis, consisting of Chapter 4, aims to characterize Hamiltonicity by means of induced subgraphs. The results in this chapter are based on the paper "Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs." Discrete Mathematics, 345(7):112869, 2022, co-authored with Joseph Cheriyan, Sepehr Hajebi, and Sophie Spirkl. We study induced subgraphs and conditions for Hamiltonicity. In particular, we characterize the minimal 2-connected non-Hamiltonian split graphs and the minimal 2-connected non-Hamiltonian triangle-free graphs

Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs

Given a graph $G$ and a graph property $P$ we say that $G$ is minimal with respect to $P$ if no proper induced subgraph of $G$ has the property $P$. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph $H$, a graph $G$ is $H$-free if $G$ has no induced subgraph isomorphic to $H$. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian $\{K_{1,3}, N_{3,1,1}\}$-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs

Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size at least $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. Let us call $(\alpha, \beta)$ \emph{triangular} if every simple edge-colored graph on $n$ vertices with at least $\alpha n$ color classes, each with at least $\beta n$ edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following: $(9/8,1/3)$ is triangular; $(1,2/5)$ is triangular. In this paper, we improve those bounds, showing the following: $(1.1077,1/3)$ is triangular; $(1,0.3988)$ is triangular. Our methods give results for infinitely many pairs $(\alpha, \beta)$, including $\beta < 1/3$; we show that $(1.3481,1/4)$ is triangular.Comment: Accepted manuscript; see DOI for journal versio

Minimal induced subgraphs of the class of 2-connected non-Hamiltonian wheel-free graphs

Given a graph G and a graph property P we say that G is minimal with respect to P if no proper induced subgraph of G has the property P. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph H, a graph G is H-free if G has no induced subgraph isomorphic to H. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian -free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs.Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-0391

Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs

The final publication is available at Elsevier via https://doi.org/10.1016/j.disc.2022.112869. © 2022. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains a claw (K1,3) or a net (a fixed six-vertex graph) as an induced subgraph. This implies that subject to being connected, these two are the only minimal (under taking induced subgraphs) graphs with no Hamiltonian path. Brousek (1998) characterized the minimal graphs that are 2-connected, non-Hamiltonian and do not contain the claw as an induced subgraph. We characterize the minimal graphs that are 2-connected and non-Hamiltonian for two classes of graphs: (1) split graphs, (2) triangle-free graphs. We remark that testing for Hamiltonicity is NP-hard in both classes.We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference numbers RGPIN-2019-04197 and RGPIN-2020-03912]

Three-Dimensional Printing of Polyaniline/Reduced Graphene Oxide Composite for High-Performance Planar Supercapacitor

We apply direct ink writing for the three-dimensional (3D) printing of polyaniline/reduced graphene oxide (PANI/RGO) composites with PANI/graphene oxide (PANI/GO) gel as printable inks. The PANI/GO gel inks for 3D printing are prepared via self-assembly of PANI and GO in a blend solvent of <i>N</i>-methyl-2-pyrrolidinone and water, and offer both shaping capability, self-sustainability, and electrical conductivity after reduction of GO. PANI/RGO interdigital electrodes are fabricated with 3D printing, and based on these electrodes, a planar solid-state supercapacitor is constructed, which exhibits high performance with an areal specific capacitance of 1329 mF cm^–2. The approach developed in this work provides a simple, economic, and effective way to fabricate PANI-based 3D architectures, which leads to promising application in future energy and electric devices at micro-nano scale