Xujun Liu

activity

20172026

most citedPacking chromatic number of subcubic graphs

2 citations · 8 across the 16 of their papers we have counts for

collaborators

Showing math.COShow all

16 papers · 1 filter

math.CO2026

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

Seog-Jin Kim, Xujun Liu, Boyan Xu

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 subcubi…

math.CO2026

Between proper and square colorings of planar graphs with maximum degree at most four

Xujun Liu, Zihui Xu, Xin Zhang

An $i$ -independent set is a vertex set whose pairwise distance is at least $i + 1$ . A proper (square) $k$ -coloring of a graph $G$ is a partition of its vertex set into $k$ independen…

math.CO2026

Every connected subcubic graph except the Petersen graph is packing $(1, 1, 2, 2)$ -colorable

Xinmin Hou, Xujun Liu, Xiangyang Wang

For a non-decreasing sequence $S = (s_{1}, s_{2}, \dots, s_{k})$ of positive integers, a packing $S$ -coloring of a graph $G$ is a partition of $V (G)$ into $V_{1}, V_{2}, \dots, V_{k}$ such t…

math.CO2026

On $(1^{2}, 2^{2})$ -packing edge-coloring of sparse subcubic graphs

Xujun Liu, Jiacheng Yang, Xin Zhang

For positive integers $ℓ$ and $k$ , a $(1^{ℓ}, 2^{k})$ -packing edge-coloring of a graph $G$ is a partition of $E (G)$ into $ℓ$ matchings and $k$ induced matchings. A graph is $d…

math.CO2025

Between proper and square colorings of sparse graphs

Ilkyoo Choi, Xujun Liu

An $i$ -independent set is a set of vertices whose pairwise distance is at least $i + 1$ . A proper coloring (resp. a square coloring) of a graph is a partition of its vertices into in…

math.CO2024

Packing edge-colorings of subcubic outerplanar graphs

Sijin Li, Yifan Li, Xujun Liu

For a sequence $S = (s_{1}, s_{2}, \dots, s_{k})$ of non-decreasing positive integers, an $S$ -packing edge-coloring (S-coloring) of a graph $G$ is a partition of $E (G)$ into $E_1, E_2,…

math.CO2024

Partition subcubic planar graphs into independent sets

Xujun Liu, Yan Wang

A packing $(1^{ℓ}, 2^{k})$ -coloring of a graph $G$ is a partition of $V (G)$ into $ℓ$ independent sets and $k$ $2$ -packings (whose pairwise vertex distance is at least $3$ ). The…

math.CO2024

Every subcubic graph is packing $(1, 1, 2, 2, 3)$ -colorable

Xujun Liu, Xin Zhang, Yanting Zhang

For a sequence $S = (s_{1}, \dots, s_{k})$ of non-decreasing integers, a packing $S$ -coloring of a graph $G$ is a partition of its vertex set $V (G)$ into $V_{1}, \dots, V_{k}$ such that fo…

math.CO2024★ 1 cited

On the $(1^{2}, 2^{4})$ -packing edge-coloring of subcubic graphs

Xujun Liu, Gexin Yu

An induced matching in a graph $G$ is a matching such that its end vertices also induce a matching. A $(1^{ℓ}, 2^{k})$ -packing edge-coloring of a graph $G$ is a partition of its e…

math.CO2022

Every subcubic multigraph is $(1, 2^{7})$ -packing edge-colorable

Xujun Liu, Michael Santana, Taylor Short

For a non-decreasing sequence $S = (s_{1}, \dots, s_{k})$ of positive integers, an $S$ -packing edge-coloring of a graph $G$ is a decomposition of edges of $G$ into disjoint sets $E_1,…

math.CO2020★ 1 cited

Packing $(1, 1, 2, 4)$ -coloring of subcubic outerplanar graphs

Alexandr Kostochka, Xujun Liu

For $1 \leq s_{1} \leq s_{2} \leq \dots \leq s_{k}$ and a graph $G$ , a packing $(s_{1}, s_{2}, \dots, s_{k})$ -coloring of $G$ is a partition of $V (G)$ into sets $V_{1}, V_{2}, \dots, V_{k}$ such that…

math.CO2019★ 1 cited

Packing $(1, 1, 2, 2)$ -coloring of some subcubic graphs

Runrun Liu, Xujun Liu, Martin Rolek +1

For a sequence of non-decreasing positive integers $S = (s_{1}, \dots, s_{k})$ , a packing $S$ -coloring is a partition of $V (G)$ into sets $V_{1}, \dots, V_{k}$ such that for each $1\leq…

math.CO2019★ 1 cited

Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

József Balogh, Alexandr Kostochka, Mikhail Lavrov +1

We solve four similar problems: For every fixed $s$ and large $n$ , we describe all values of $n_{1}, \dots, n_{s}$ such that for every $2$ -edge-coloring of the complete $s$ -partite grap…

math.CO2019

Ramsey numbers of ordered graphs under graph operations

Jesse Geneson, Amber Holmes, Xujun Liu +3

An ordered graph $G$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $G$ is the smallest integer $N$ such that…

math.CO2017★ 2 cited

Cubic graphs with small independence ratio

József Balogh, Alexandr Kostochka, Xujun Liu

Let $i (r, g)$ denote the infimum of the ratio $\frac{α ( G )}{∣ V ( G ) ∣}$ over the $r$ -regular graphs of girth at least $g$ , where $α (G)$ is the independence number of $G$ , and let $i(r,\…

math.CO2017★ 2 cited

Packing chromatic number of subcubic graphs

József Balogh, Alexandr Kostochka, Xujun Liu

A packing $k$ -coloring of a graph $G$ is a partition of $V (G)$ into sets $V_{1}, \dots, V_{k}$ such that for each $1 \leq i \leq k$ the distance between any two distinct $x, y \in V_{i}$ is a…

On (1,24) and (1,25)-packing edge-coloring of sparse subcubic graphs

Between proper and square colorings of planar graphs with maximum degree at most four

Every connected subcubic graph except the Petersen graph is packing (1,1,2,2)-colorable

On (12,22)-packing edge-coloring of sparse subcubic graphs

Between proper and square colorings of sparse graphs

Packing edge-colorings of subcubic outerplanar graphs

Partition subcubic planar graphs into independent sets

Every subcubic graph is packing (1,1,2,2,3)-colorable

On the (12,24)-packing edge-coloring of subcubic graphs

Every subcubic multigraph is (1,27)-packing edge-colorable

Packing (1,1,2,4)-coloring of subcubic outerplanar graphs

Packing (1,1,2,2)-coloring of some subcubic graphs

Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

Ramsey numbers of ordered graphs under graph operations

Cubic graphs with small independence ratio

Packing chromatic number of subcubic graphs

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

Every connected subcubic graph except the Petersen graph is packing $(1, 1, 2, 2)$ -colorable

On $(1^{2}, 2^{2})$ -packing edge-coloring of sparse subcubic graphs

Every subcubic graph is packing $(1, 1, 2, 2, 3)$ -colorable

On the $(1^{2}, 2^{4})$ -packing edge-coloring of subcubic graphs

Every subcubic multigraph is $(1, 2^{7})$ -packing edge-colorable

Packing $(1, 1, 2, 4)$ -coloring of subcubic outerplanar graphs

Packing $(1, 1, 2, 2)$ -coloring of some subcubic graphs