arXiv subjects

Jaehoon Kim

Publications and source records attributed to Jaehoon Kim.

88 records · Page 5Linked to original sources

arXiv2018

Dynamic choosability of triangle-free graphs and sparse random graphs

The \textit{$r$-dynamic choosability} of a graph $G$, written ${\rm ch}_r(G)$, is the least $k$ such that whenever each vertex is assigned a list of at least $k$ colors a proper coloring can be chosen from the lists so that every vertex $v$ has at least $\min\{d_G(v),r\}$ neighbors of distinct colors. Let ${\rm ch}(G)$ denote the choice number of $G$. In this paper, we prove ${\rm ch}_r(G)\leq (1+o(1)){\rm ch}(G)$ when $\frac{Δ(G)}{δ(G)}$ is bounded. We also show that there exists a constant $C$ such that for the random graph $G=G(n,p)$ with $\frac{2}{n}<p\leq \frac{1}{2}$, it holds that ${\rm ch}_2(G)\leq {\rm ch}(G) + C$, asymptotically almost surely. Also if $G$ is triangle-free regualr graph, then ${\rm ch}_2(G)\leq {\rm ch}(G)+86$ holds.

math.CO↗

arXiv2018

Strong edge-colorings of sparse graphs with large maximum degree

A {\em strong $k$-edge-coloring} of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The {\em strong chromatic index} $χ_s'(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ admits a strong $k$-edge-coloring. We give bounds on $χ_s'(G)$ in terms of the maximum degree $Δ(G)$ of a graph $G$. when $G$ is sparse, namely, when $G$ is $2$-degenerate or when the maximum average degree ${\rm Mad}(G)$ is small. We prove that the strong chromatic index of each $2$-degenerate graph $G$ is at most $5Δ(G) +1$. Furthermore, we show that for a graph $G$, if ${\rm Mad}(G)< 8/3$ and $Δ(G)\geq 9$, then $χ_s'(G)\leq 3Δ(G) -3$ (the bound $3Δ(G) -3$ is sharp) and if ${\rm Mad}(G)<3$ and $Δ(G)\geq 7$, then $χ_s'(G)\leq 3Δ(G)$ (the restriction ${\rm Mad}(G)<3$ is sharp).

math.CO↗

arXiv2018

Two-regular subgraphs of odd-uniform hypergraphs

Let $k\ge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $\binom{n-1}{k-1} + \lfloor\frac{n-1}{k} \rfloor$, and the equality holds if and only if $H$ is a full $k$-star with center $v$ together with a maximal matching omitting $v$. This verifies a conjecture of Mubayi and Verstraëte.

math.CO↗

arXiv2018

Sparse spanning $k$-connected subgraphs in tournaments

In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs.

math.CO↗

arXiv2017

A blow-up lemma for approximate decompositions

We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex graph and suppose $H_1,\dots,H_s$ are bounded degree $n$-vertex graphs with $\sum_{i=1}^{s} e(H_i) \leq (1-o(1)) e(G)$. Then $H_1,\dots,H_s$ can be packed edge-disjointly into $G$. The case when $G$ is the complete graph $K_n$ implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi's regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy and Szemerédi to the setting of approximate decompositions.

math.CO↗

arXiv2017

Catalyst design using actively learned machine with non-ab initio input features towards CO2 reduction reactions

In conventional chemisorption model, the d-band center theory (augmented sometimes with the upper edge of d-band for imporved accuarcy) plays a central role in predicting adsorption energies and catalytic activity as a function of d-band center of the solid surfaces, but it requires density functional calculations that can be quite costly for large scale screening purposes of materials. In this work, we propose to use the d-band width of the muffin-tin orbital theory (to account for local coordination environment) plus electronegativity (to account for adsorbate renormalization) as a simple set of alternative descriptors for chemisorption, which do not demand the ab initio calculations. This pair of descriptors are then combined with machine learning methods, namely, artificial neural network (ANN) and kernel ridge regression (KRR), to allow large scale materials screenings. We show, for a toy set of 263 alloy systems, that the CO adsorption energy can be predicted with a remarkably small mean absolute deviation error of 0.05 eV, a significantly improved result as compared to 0.13 eV obtained with descriptors including costly d-band center calculations in literature. We achieved this high accuracy by utilizing an active learning algorithm, without which the accuracy was 0.18 eV otherwise. As a practical application of this machine, we identified Cu3Y@Cu as a highly active and cost-effective electrochemical CO2 reduction catalyst to produce CO with the overpotential 0.37 V lower than Au catalyst.

cond-mat.mtrl-sci↗

arXiv2017

Proof of Komlós's conjecture on Hamiltonian subsets

Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.

math.CO↗

arXiv2016

Regular subgraphs of uniform hypergraphs

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if $r\in\{3,4\}$, $r\mid k$ and $k,n$ are both sufficiently large, then the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $r$-regular subgraphs is exactly ${{n-1} \choose {k-1}}$, with equality only if all edges contain a specific vertex $v$. We also ask some related questions.

math.CO↗

arXiv2015

The difference and ratio of the fractional matching number and the matching number of graphs

Given a graph $G$, the matching number of $G$, written $α'(G)$, is the maximum size of a matching in $G$, and the fractional matching number of $G$, written $α'_f(G)$, is the maximum size of a fractional matching of $G$. In this paper, we prove that if $G$ is an $n$-vertex connected graph that is neither $K_1$ nor $K_3$, then $α'_f(G)-α'(G) \le \frac{n-2}6$ and $\frac{α'_f(G)}{α'(G)} \le \frac{3n}{2n+2}$. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.

math.CO↗

arXiv2015

A relative of Hadwiger's conjecture

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.

math.CO↗

arXiv2015

Forbidding induced even cycles in a graph: typical structure and counting

We determine, for all $k\geq 6$, the typical structure of graphs that do not contain an induced $2k$-cycle. This verifies a conjecture of Balogh and Butterfield. Surprisingly, the typical structure of such graphs is richer than that encountered in related results. The approach we take also yields an approximate result on the typical structure of graphs without an induced $8$-cycle or without an induced $10$-cycle.

math.CO↗

arXiv2015

Bipartitions of highly connected tournaments

We show that if $T$ is a strongly $10^9k^6\log(2k)$-connected tournament, there exists a partition $A, B$ of $V(T)$ such that each of $T[A]$, $T[B]$ and $T[A,B]$ is strongly $k$-connected. This provides tournament analogues of two partition conjectures of Thomassen regarding highly connected graphs.

math.CO↗

arXiv2014

On the Erdos-Ko-Rado Theorem and the Bollobas Theorem for t-intersecting families

A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t} \choose {k-t}}$ if $n\geq n_0(k,t)$. Alon, Aydinian, and Huang considered families generalizing intersecting families, and proved the same bound. In this paper, we give a strengthening of their result by considering families generalizing $t$-intersecting families for all $t \geq 1$. In 2004, Talbot generalized Bollobás's Two Families Theorem to $t$-intersecting families. In this paper, we proved a slight generalization of Talbot's result by using the probabilistic method.

math.CO↗

arXiv2014

On the number of $r$-matchings in a Tree

An $r$-matching in a graph $G$ is a collection of edges in $G$ such that the distance between any two edges is at least $r$. A $2$-matching is also called an induced matching. In this paper, we estimate the maximum number of $r$-matchings in a tree of fixed order. We also prove that the $n$-vertex path has the maximum number of induced matchings among all $n$-vertex trees.

math.CO↗

arXiv2012

Harmonious Coloring of Trees with Large Maximum Degree

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a harmonious coloring of $G$. We show that if $T$ is a forest of order $n$ with maximum degree $Δ(T)\geq \frac{n+2}{3}$, then $$h(T)= Δ(T)+2, & if $T$ has non-adjacent vertices of degree $Δ(T)$; Δ(T)+1, & otherwise. $$ Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.

math.CO↗

arXiv2011

New results in $t$-tone coloring of graphs

A $t$-tone $k$-coloring of $G$ assigns to each vertex of $G$ a set of $t$ colors from $\{1,..., k\}$ so that vertices at distance $d$ share fewer than $d$ common colors. The {\it $t$-tone chromatic number} of $G$, denoted $τ_t(G)$, is the minimum $k$ such that $G$ has a $t$-tone $k$-coloring. Bickle and Phillips showed that always $τ_2(G) \le [Δ(G)]^2 + Δ(G)$, but conjectured that in fact $τ_2(G) \le 2Δ(G) + 2$; we confirm this conjecture when $Δ(G) \le 3$ and also show that always $τ_2(G) \le \ceil{(2 + \sqrt{2})Δ(G)}$. For general $t$ we prove that $τ_t(G) \le (t^2+t)Δ(G)$. Finally, for each $t\ge 2$ we show that there exist constants $c_1$ and $c_2$ such that for every tree $T$ we have $c_1 \sqrt{Δ(T)} \le τ_t(T) \le c_2\sqrt{Δ(T)}$.

math.CO↗