arXiv subjects

Seog-Jin Kim

Publications and source records attributed to Seog-Jin Kim.

At least 19 recordsLinked to original sources

arXiv2026

On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs

A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of the edge set $E(G)$ into $j$ matchings and $k$ induced matchings. Hocquard, Lajou, and Lu\v zar found a subcubic planar graph of girth $3$ that has no $(1,2^5)$-packing edge-coloring and also conjectured that every subcubic planar graph has a $(1,2^6)$-packing edge-coloring. We also notice that for every fixed positive integer $k$ there exists a subcubic planar graph with girth $k$ that is not $(1,2^3)$-packing edge-colorable. It is natural to consider what is the minimum positive integer $k_1$ such that every subcubic planar graph with girth at least $k_1$ is $(1,2^5)$-packing edge-colorable. Furthermore, we also consider what is the minimum positive integer $k_2$ such that every subcubic planar graph with girth at least $k_2$ is $(1,2^4)$-packing edge-colorable. In this paper, we show both $k_1$ and $k_2$ are finite, and in fact $5 \le k_1 \le 12$ and $6 \le k_2 \le 16$.

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arXiv2026

The Alon-Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$

The Alon--Tarsi number $AT(G)$ of a graph $G$, defined via the graph polynomial, is a strengthening of the list chromatic number $χ_{\ell}(G)$. We study the Alon--Tarsi number of squares of planar graphs. The square of a graph $G$ is the graph obtained by joining every pair of vertices whose distance in $G$ is at most $2$. Recently, Kim and Luo (2026) proved that $χ_{\ell}(G^2)\le 6$ for every subcubic planar graph containing no $k$-cycles for $4\le k\le 8$. We strengthen this result by proving that $AT(G^2)\le 6$ for every such graph $G$.

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arXiv2025

The square of a subcubic planar graph without a 5-cycle is 7-choosable

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen [12] showed that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently Kim and Lian [11] showed that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].

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arXiv2025

Squares of subcubic planar graphs without cycles of length 4-8 are 6-choosable

The {\em square} of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and an edge between any two vertices at distance at most $2$ in $G$. Wegner (1977) conjectured that for a planar graph $G$, $χ(G^2) \leq 7$ if $Δ(G) = 3$, $χ(G^2) \leq Δ(G)+5$ if $4 \leq Δ(G) \leq 7$, and $χ(G^2) \leq \lfloor 3Δ(G)/2 \rfloor$ if $Δ(G) \geq 8$, and Thomassen (2018) confirmed the conjecture for $Δ(G) = 3$. Dvořák et al. (2008) and Feder et al. (2021) further conjectured that $χ(G^2) \leq 6$ for cubic bipartite planar graphs. A natural question is whether this bound also holds for the list-chromatic number, i.e., whether $χ_{\ell}(G^2) \leq 6$ for such graphs. More generally, it is of interest to determine sufficient conditions ensuring $χ_{\ell}(G^2) \leq 6$ for subcubic planar graphs. In this paper, we prove that $χ_{\ell}(G^2) \leq 6$ for subcubic planar graphs containing no $k$-cycles for $4 \leq k \leq 8$, improving a result of Cranston and Kim (2008).

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arXiv2025

The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable

The square of a graph $G$, denoted by $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen (2018) and independently, Hartke, Jahanbekam and Thomas (2016) proved that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently, Kim and Lian (2024) proved that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. In this paper, we prove that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles, which improves the result of Kim and Lian.

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arXiv2023

Tight upper bound on the clique size in the square of 2-degenerate graphs

The {\em square} of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. In general, $Δ(G) + 1 \leq χ(G^2) \leq Δ(G)^2 +1$ for every graph $G$. Charpentier [1] asked whether $χ(G^2) \leq 2 Δ(G)$ if $mad(G) < 4$. But Hocquard, Kim, and Pierron [6] answered his question negatively. For every even value of $Δ(G)$, they constructed a 2-degenerate graph $G$ such that $ω(G^2) = \frac{5}{2} Δ(G)$. Note that if $G$ is a 2-degenerate graph, then $mad(G) < 4$. Thus, we have that \[ {\displaystyle \frac{5}{2} Δ(G) \leq \max \{χ(G^2) : G \mbox{ is a 2-degenerate graph} \} \leq 3 Δ(G) +1}. \] So, it was naturally asked whether there exists a constant $D_0$ such that $χ(G^2) \leq \frac{5}{2} Δ(G)$ if $G$ is a 2-degenerate graph with $Δ(G) \geq D_0$. Recently Cranston and Yu [3] showed that $ω(G^2) \leq \frac{5}{2} Δ(G)+72$ if $G$ is a 2-degenerate graph, and $ω(G^2) \leq \frac{5}{2} Δ(G)+60$ if $G$ is a 2-degenerate graph with $Δ(G) \geq 1729$. We show that there exists a constant $D_0$ such that $ω(G^2) \leq \frac{5}{2} Δ(G)$ if $G$ is a 2-degenerate graph with $Δ(G) \geq D_0$. This upper bound on $ω(G^2)$ is tight by the construction in [6].

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arXiv2023

The square of every subcubic planar graph of girth at least 6 is 7-choosable

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen (2018) and Hartke, Jahanbekam and Thomas (2016) proved that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Cranston and Kim (2008) showed that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6.

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arXiv2020

$K_{r+1}$-saturated graphs with small spectral radius

For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $n\to\infty$.

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arXiv2019

The Alon-Tarsi number of $K_5$-minor-free graphs

In this paper, we show the following three theorems. Let $G$ be a $K_5$-minor-free graph. Then Alon-Tarsi number of $G$ is at most $5$, there exists a matching $M$ of $G$ such that the Alon-Tarsi number of $G-M$ is at most $4$, and there exists a forest $F$ such that the Alon-Tarsi number of $G-E(F)$ is at most $3$.

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arXiv2019

Planar graphs without 7-cycles and butterflies are DP-4-colorable

DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvořák and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP-$4$-colorable. In particular, for each $k \in \{3, 4, 5, 6\}$, every planar graph without $k$-cycles is DP-$4$-colorable. In this paper, we prove that every planar graph without $7$-cycles and butterflies is DP-$4$-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters formed by many triangles.

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arXiv2019

On list 3-dynamic coloring of near-triangulations

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r, deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $χ_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list $r$-dynamic chromatic number of a graph $G$ is denoted by $ch_r^d(G)$. Loeb et al. $[11]$ showed that $ch_3^d(G)\leq 10$ for every planar graph $G$, and there is a planar graph $G$ with $χ_3^d(G)= 7$. In this paper, we study a special class of planar graphs which have better upper bounds of $ch_3^d(G)$. We prove that $ch_3^d(G) \leq 6$ if $G$ is a planar graph which is near-triangulation, where a near-triangulation is a planar graph whose bounded faces are all 3-cycles.

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arXiv2019

The Alon-Tarsi number of subgraphs of a planar graph

This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $Δ(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable.

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arXiv2019

Coloring squares of graphs with mad constraints

A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is the graph defined by $V(G)=V(G^2)$ and $uv \in E(G^2)$ if and only if the distance between $u$ and $v$ is at most two. We denote by $χ(G^2)$ the chromatic number of $G^2$, which is the least integer $k$ such that a $k$-coloring of $G^2$ exists. By definition, at least $Δ(G)+1$ colors are needed for this goal, where $Δ(G)$ denotes the maximum degree of the graph $G$. In this paper, we prove that the square of every graph $G$ with $\text{mad}(G)<4$ and $Δ(G) \geqslant 8$ is $(3Δ(G)+1)$-choosable and even correspondence-colorable. Furthermore, we show a family of $2$-degenerate graphs $G$ with $\text{mad}(G)<4$, arbitrarily large maximum degree, and $χ(G^2)\geqslant \frac{5Δ(G)}{2}$, improving the result of Kim and Park.

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arXiv2017

Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable

DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which is an extension of the two results above.

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arXiv2017

A note on a Brooks' type theorem for DP-coloring

Dvořák and Postle \cite{DP} introduced a \textit{DP-coloring} of a simple graph as a generalization of a list-coloring. They proved a Brooks' type theorem for a DP-coloring, and Bernshteyn, Kostochka and Pron \cite{BKP} extended it to a DP-coloring of multigraphs. However, detailed structure when a multigraph does not admit a DP-coloring was not specified in \cite{BKP}. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.

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arXiv2017

A Sufficient condition for DP-4-colorability

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each $k \in \{3, 4, 5, 6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results.

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arXiv2017

List 3-dynamic coloring of graphs with small maximum average degree

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r,°_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $χ_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The {\em list $r$-dynamic chromatic number} of a graph $G$ is denoted by $ch_r^d(G)$. Recently, Loeb et al. [UI] showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. [Lai-16] studied the maximum average condition to have $χ_3^d (G) \leq 4, \ 5$, or $6$. On the other hand, Song et al. [SLW] showed that if $G$ is planar with girth at least 6, then $χ_r^d(G)\le r+5$ for any $r\ge 3$. In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $ch^d_3(G) \leq 6$ if $mad(G) < \frac{18}{7}$, $ch^d_3(G) \leq 7$ if $mad(G) < \frac{14}{5}$, and $ch^d_3(G) \leq 8$ if $mad(G) < 3$. All of the bounds are tight.

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arXiv2016

Cycles with two blocks in $k$-chromatic digraphs

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $c(k,\ell)$ is an oriented cycle which consists of two internally (vertex) disjoint directed paths of lengths at least $k$ and $\ell$, respectively, from a vertex to another one. A problem of Addario-Berry, Havet and Thomassé (2007) asked if, given positive integers $k$ and $\ell$ such that $k+\ell\ge 4$, any strongly connected digraph $D$ containing no $c(k,\ell)$ has chromatic number at most $k+\ell-1$. In this paper, we show that such digraph $D$ has chromatic number at most $O((k+\ell)^2)$, improving the previous upper bound $O((k+\ell)^4)$ obtained by Cohen, Havet, Lochet and Nisse (2016). In fact, we are able to find a digraph which shows that the answer to the above problem is no. We also show that if in addition $D$ is Hamiltonian, then its underlying simple graph is $(k+\ell-1)$-degenerate and thus the chromatic number of $D$ is at most $k+\ell$, which is tight.

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