Between proper and strong edge-coloring of graphs with maximum degree at most four

Aistis Atminas; Xujun Liu; Zihui Xu

Between proper and strong edge-coloring of graphs with maximum degree at most four

Aistis Atminas
, Xujun Liu^*
, Zihui Xu

^*Corresponding author for this work

Research output: Other contribution

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Abstract

A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of its edge set $E(G)$ into $j$ matchings and $k$ induced matchings. It can be viewed as intermediate colorings between proper and strong edge-coloring. The study of this topic on subcubic graphs has drawn considerable attention recently, with many conjectures and open problems proposed and resolved. However, very few results are known for graphs with maximum degree four or more. Henderson, Kennedy, and Santana showed that every graph with degree at most four has a $(1,2^{19})$-packing edge-coloring and a $(1^2,2^{17})$-packing edge-coloring.

In this paper, we improve the results of Henderson, Kennedy, and Santana by showing every graph with degree at most four has a $(1,2^{17})$-packing edge-coloring. As a corollary of our result, every graph with degree at most four has a $(1^2,2^{16})$-packing edge-coloring.

Original language	English
Number of pages	10
Publication status	Submitted - 31 Aug 2026

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Packing_edge_coloring_17Submitted manuscript, 128 KBLicence: CC BY-NC-ND

Cite this

@misc{6693544959104c91b9a286d61bce8dd1,

title = "Between proper and strong edge-coloring of graphs with maximum degree at most four",

abstract = "A \$(1\textasciicircum{}j,2\textasciicircum{}k)\$-packing edge-coloring of a graph \$G\$ is a partition of its edge set \$E(G)\$ into \$j\$ matchings and \$k\$ induced matchings. It can be viewed as intermediate colorings between proper and strong edge-coloring. The study of this topic on subcubic graphs has drawn considerable attention recently, with many conjectures and open problems proposed and resolved. However, very few results are known for graphs with maximum degree four or more. Henderson, Kennedy, and Santana showed that every graph with degree at most four has a \$(1,2\textasciicircum{}\{19\})\$-packing edge-coloring and a \$(1\textasciicircum{}2,2\textasciicircum{}\{17\})\$-packing edge-coloring. In this paper, we improve the results of Henderson, Kennedy, and Santana by showing every graph with degree at most four has a \$(1,2\textasciicircum{}\{17\})\$-packing edge-coloring. As a corollary of our result, every graph with degree at most four has a \$(1\textasciicircum{}2,2\textasciicircum{}\{16\})\$-packing edge-coloring.",

author = "Aistis Atminas and Xujun Liu and Zihui Xu",

year = "2026",

month = aug,

day = "31",

language = "English",

type = "Other",

}

TY - GEN

T1 - Between proper and strong edge-coloring of graphs with maximum degree at most four

AU - Atminas, Aistis

AU - Liu, Xujun

AU - Xu, Zihui

PY - 2026/8/31

Y1 - 2026/8/31

N2 - A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of its edge set $E(G)$ into $j$ matchings and $k$ induced matchings. It can be viewed as intermediate colorings between proper and strong edge-coloring. The study of this topic on subcubic graphs has drawn considerable attention recently, with many conjectures and open problems proposed and resolved. However, very few results are known for graphs with maximum degree four or more. Henderson, Kennedy, and Santana showed that every graph with degree at most four has a $(1,2^{19})$-packing edge-coloring and a $(1^2,2^{17})$-packing edge-coloring. In this paper, we improve the results of Henderson, Kennedy, and Santana by showing every graph with degree at most four has a $(1,2^{17})$-packing edge-coloring. As a corollary of our result, every graph with degree at most four has a $(1^2,2^{16})$-packing edge-coloring.

AB - A $(1^j,2^k)$-packing edge-coloring of a graph $G$ is a partition of its edge set $E(G)$ into $j$ matchings and $k$ induced matchings. It can be viewed as intermediate colorings between proper and strong edge-coloring. The study of this topic on subcubic graphs has drawn considerable attention recently, with many conjectures and open problems proposed and resolved. However, very few results are known for graphs with maximum degree four or more. Henderson, Kennedy, and Santana showed that every graph with degree at most four has a $(1,2^{19})$-packing edge-coloring and a $(1^2,2^{17})$-packing edge-coloring. In this paper, we improve the results of Henderson, Kennedy, and Santana by showing every graph with degree at most four has a $(1,2^{17})$-packing edge-coloring. As a corollary of our result, every graph with degree at most four has a $(1^2,2^{16})$-packing edge-coloring.

M3 - Other contribution

ER -