Discrete Mathematics最新文献

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Hamiltonian decompositions of the wreath product of hamiltonian decomposable digraphs 哈密顿可分解有向图环积的哈密顿分解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-27 DOI: 10.1016/j.disc.2026.115012

Alice Lacaze-Masmonteil

We confirm most open cases of a conjecture that first appeared in Alspach et al. (1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable. Specifically, we show that the wreath product of a hamiltonian decomposable directed graph G, such that

| V (G) |

is even and

| V (G) | ⩾ 2

, with a hamiltonian decomposable directed graph H, such that

| V (H) | ⩾ 4

, is also hamiltonian decomposable except possibly when G is a directed cycle and H is a directed graph of odd order that admits a decomposition into c directed hamiltonian cycle where c is odd and

3 ⩽ c ⩽ | V (H) | - 2

我们证实了Alspach等人（1987）首次提出的一个猜想的大多数开放情况，该猜想规定两个哈密顿可分解有向图的环（字典）积也是哈密顿可分解的。具体地说，我们表明哈密顿可分解有向图G的环积，使得|V(G)|是偶数，|V(G)|小于2，具有哈密顿可分解有向图H，使得|V(H)|小于4，也是哈密顿可分解的，除非可能当G是有向循环并且H是奇阶有向图允许分解成c有向哈密顿循环其中c是奇数并且3≤c≤|V(H)|−2。

引用次数: 0

Spectral Turán-type problems on sparse spanning graphs 稀疏生成图上的谱Turán-type问题

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-27 DOI: 10.1016/j.disc.2026.115016

Lele Liu , Bo Ning

<div><div>Let F be a graph, and let <math><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math>, <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math>, and <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> denote the classes of graphs that attain, respectively, the maximum number of edges, the maximum adjacency spectral radius, and the maximum signless Laplacian spectral radius over all n-vertex graphs that do not contain F as a subgraph. A fundamental problem in spectral extremal graph theory is to characterize all graphs F for which <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> when n is sufficiently large. Establishing the conjecture of Cioabă et al. (2022) [10], Wang et al. (2023) [54] proved that: for any graph F such that the graphs in <math><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> are Turán graphs plus <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> edges, <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> for sufficiently large n. In addition, another interesting problem in spectral extremal graph theory is to characterize all graphs F such that <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> for sufficiently large n.</div><div>In this paper, we give new contribution to the problems mentioned above. First, we present a substantial collection of examples of graphs F for which <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> holds when n is sufficiently large, focusing on n-vertex graph F with no isolated vertices and maximum degree <math><mi>Δ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>/</mo><mn>40</mn></math>. Second, under the same conditions on F, we prove that <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi

设F是一个图，设EX（n，F）、SPEXA（n，F）和SPEXQ（n，F）分别表示在所有不包含F作为子图的n顶点图上获得最大边数、最大邻接谱半径和最大无符号拉普拉斯谱半径的图的类别。谱极值图论中的一个基本问题是，当n足够大时，对所有的图F （SPEXA(n，F)）进行描述。通过建立cioabu et al. (2022) b[10]的猜想，Wang et al. (2023) b[54]证明：对于任意图F，使得EX（n，F）中的图为Turán图加O(1)条边，对于足够大的n， SPEXA（n，F）的任任任任，对于谱极值图论中的另一个有趣问题是，对所有图F进行描述，使得对于足够大的n， SPEXA(n，F)=SPEXQ（n，F）。本文对上述问题给出了新的贡献。首先，我们给出了大量在n足够大时，SPEXA(n，F)≥≥EX（n，F）成立的图F的例子，重点关注无孤立顶点且最大度数Δ(F)≤n/40的n顶点图F。其次，在F上相同的条件下，我们证明了对于足够大的n， SPEXA(n，F)=SPEXQ（n，F）。这些结果可以看作是Alon和Yuster(2013)[1]定理的谱类比。进一步，作为直接推论，我们得到了几类特殊图存在的紧谱条件，包括团因子、汉密尔顿环的k次幂和图中的k因子。第一类特殊的图给出了Feng的一个问题的正答案，第二类特殊的图扩展了Yan等人之前的结果。

引用次数: 0

Weak saturation numbers for the union of disjoint graphs 不相交图并集的弱饱和数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-27 DOI: 10.1016/j.disc.2026.115011

Yu Zhang, Rong-Xia Hao, Zhen He, Jianbing Liu

<div><div>Let F and H be two graphs. A spanning subgraph G of F is said to be weakly <math><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo></math>-saturated if there exists an ordering <math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>t</mi></mrow></msub></math> of the edges in <math><mi>E</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>∖</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> such that, for each <math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>t</mi><mo>]</mo></math>, the addition of <math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math> to <math><mi>G</mi><mo>+</mo><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo></math> creates a new copy of H containing <math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math>. The weak saturation number of H with respect to F is defined as <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>:</mo><mi>G</mi><mtext> is weakly </mtext><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo><mtext>-saturated</mtext><mo>}</mo></math>. Kronenberg et al. (2021) [7] determined the exact values of <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math> and <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math>. In this paper, we generalize previous results by determining the exact values of <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math> and <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math> for <math><mi>r</mi><mo>≥</mo><mn>1</mn></math>, and provide both upper and lower bounds for <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></math>. Additionally, we determine <math><mi>w</mi><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><munderover><mo>⋃</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mr

设F和H是两个图。F的生成子图G是弱（F，H）饱和的，如果存在E(F)∈E(G)中的边的排序e1，…，et，使得对于每一个i∈[t]， ei加上G+{e1，…，ei−1}产生一个包含ei的H的新副本。H相对于F的弱饱和数定义为wsat(F，H)=min (|E(G)|:G是弱（F，H）饱和}。Kronenberg et al.(2021)[7]确定了wsat（n,Kt,t）和wsat（n,Kt,t+1）的确切值。本文通过确定r≥1时wsat（n,rKt,t）和wsat（n,rKt,t+1）的精确值，推广了前人的结果，并给出了wsat（n,rKs,t）的上界和下界。此外，我们确定了不相交完全图并集的wsat(n,∈i=1qKti)，这改进了Faudree等人关于wsat（n,qKt）的已知结果。

引用次数: 0

Deletable edges in 3-connected graphs and their applications 3连通图中的可删除边及其应用

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-23 DOI: 10.1016/j.disc.2026.114994

S.R. Kingan

<div><div>Let G and H be simple 3-connected graphs such that G has an H-minor. An edge e in G is called H-deletable if <math><mi>G</mi><mo>﹨</mo><mi>e</mi></math> is 3-connected and has an H-minor. The main result in this paper establishes that, if G has no H-deletable edges, then there exists a sequence of simple 3-connected graphs <math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math> with no H-deletable edges such that <math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≅</mo><mi>H</mi></math>, <math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>G</mi></math>, and for <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math> one of three possibilities holds: <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><mi>f</mi></math>; <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><mi>f</mi><mo>﹨</mo><mi>e</mi></math> where e and f are incident to a degree 3 vertex in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math>; or <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><mi>w</mi></math> where w is a degree 3 vertex in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math>. We give several applications including a graph-theoretic proof of the matroid theory result known as the Strong Splitter Theorem, a short new proof of Dirac's characterization of 3-connected graphs with no minor isomorphic to the prism graph, and an extension of a result by Halin that bounds the number of edges in a minimally 3-connected graph. Halin proved that if G is a minimally 3-connected graph on <math><mi>n</mi><mo>≥</mo><mn>8</mn></math> vertices, then <math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>9</mn></math> and equality holds if and only if <math><mi>G</mi><mo>≅</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></math>. We give a different proof of Halin's result and extend it by identifying the minimally 3-connected infinite family of graphs with <math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>10</mn></math>. Finally, we extend the main theorem to mat

设G和H是简单的3连通图，其中G有一个H小调。G中的边e被称为h可删除边，如果Ge是3连通的并且有一个h小调。本文的主要结果证明，如果G没有H-可删除边，则存在一个无H-可删除边的简单3连通图序列G0，…，Gk，使得G0 = H, Gk=G，且对于1≤i≤k，有三种可能之一成立：Gi−1=Gi/f；Gi−1=Gi/fe，其中e和f入射到Gi中的3次顶点；或者Gi−1=Gi−w，其中w是Gi中的一个3度顶点。我们给出了几个应用，包括矩阵理论结果强分裂定理的图论证明，Dirac关于无小同构棱镜图的3连通图刻划的一个简短的新证明，以及Halin关于最小3连通图边数限定的一个结果的推广。Halin证明了如果G是n≥8个顶点上的最小3连通图，则|E(G)|≤3n−9，且等式成立当且仅当G = K3,n−3。我们给出了Halin结果的另一种证明，并通过识别|E(G)|=3n−10的最小3连通无限族图来扩展它。最后，将主要定理推广到拟阵中。

引用次数: 0

Some frustrating questions on dimensions of products of posets 关于偏序集积维数的一些令人沮丧的问题

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-21 DOI: 10.1016/j.disc.2026.115002

George M. Bergman

The definition of the dimension of a poset is recalled. For a subposet P of a direct product of

d > 0

chains, and an integer

n > 0

, a condition is developed which implies that for any family of n chains

{(T_{j})}_{j \in n}

, one has

\dim (P \times \prod_{j \in n} T_{j}) \leq d

. Applications are noted.

Open questions, old and new, on dimensions of product posets are stated, and some other numerical invariants of posets that seem useful for studying these questions are developed. Some variants of the concept of the dimension of a poset from the literature are also recalled.

In a final section, independent of the other results, it is noted that by the compactness theorem of first-order logic, an infinite poset P has finite dimension d if and only if d is the supremum of the dimensions of its finite subposets.

回顾偏序集维数的定义。对于d>；0链的直积P与整数n>；0，给出了一个条件，该条件表明对于任意n个链（Tj）j∈n的族，有dim (px∏j∈nTj)≤d。应用程序被记录。提出了关于积序集维数的新老开放问题，并给出了对研究这些问题有用的其它一些积序集的数值不变量。还回顾了文献中偏序集维数概念的一些变体。在最后一节中，独立于其他结果，我们注意到，通过一阶逻辑的紧性定理，当且仅当d是其有限集合维数的上限时，无限偏集P具有有限维d。

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引用次数: 0

The Kneser chromatic function distinguishes trees 克奈瑟色函数区分树

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-20 DOI: 10.1016/j.disc.2026.115009

Yusaku Nishimura

R.P. Stanley defined an invariant for graphs called the chromatic symmetric function and conjectured it is a complete invariant for trees. Miezaki et al. generalized the chromatic symmetric function and defined the Kneser chromatic functions denoted by

{X_{K_{N, k}}}_{k \in N}

, and rephrase Stanley's conjecture that

X_{K_{N, 1}}

is a complete invariant for trees. This paper shows

X_{K_{N, 2}}

is a complete invariant for trees.

R.P. Stanley定义了一个图的不变量，称为色对称函数，并推测它是树的完全不变量。miiezaki等人推广了色对称函数，定义了Kneser色函数{XKN，k}k∈N，并重新表述了Stanley关于XKN，1是树的完全不变量的猜想。本文证明XKN，2是树的完全不变量。

引用次数: 0

Around the number of trees in distance-hereditary graphs 关于距离遗传图中树的数目

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-20 DOI: 10.1016/j.disc.2026.115010

Danila Cherkashin , Pavel Prozorov

Counting the number of spanning trees in specific classes of graphs has attracted growing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained during the 2020s. Our main approach is to study the behavior of the vertex (degree) enumerator polynomial of a distance-hereditary graph under certain graph-theoretical operations. The first result provides a factorization formula applicable to graphs admitting a cut whose edges form a complete bipartite subgraph.

One of the central open problems in this area is Ehrenborg's conjecture, which asserts that a Ferrers–Young graph maximizes the number of spanning trees among all bipartite graphs with the same degree sequence. The second main result of this paper shows the equivalence between Ehrenborg's conjecture and its polynomial version.

近年来，计算特定图类中生成树的数量引起了越来越多的关注。在本文中，我们对本世纪20年代获得的几个结果进行了统一的证明和推广。我们的主要方法是研究距离遗传图的顶点（度）枚举数多项式在某些图论操作下的行为。第一个结果提供了一个适用于有切点的图的因式分解公式，这些切点的边构成了完全二部子图。该领域的一个中心开放问题是Ehrenborg猜想，它断言ferers - young图在所有具有相同次序列的二部图中生成树的数量最大。本文的第二个主要结果证明了Ehrenborg猜想与其多项式版本之间的等价性。

引用次数: 0

A Metzler matrix of a group covering of a digraph 有向图的群覆盖的Metzler矩阵

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.disc.2026.115006

Yusuke Ide , Takashi Komatsu , Norio Konno , Iwao Sato

We present a decomposition formula for the determinant of a Metzler matrix

A (H)

of a group covering H of a digraph D. Furthermore, we introduce an L-function of D with respect to its Metzler matrix

A (D)

, and present a determinant expression of it. As a corollary, we present a decomposition formula for the determinant of a Metzler matrix

A (H)

of a group covering H of D by its L-functions.

给出了有向图D中覆盖H的群的Metzler矩阵a (H)的行列式的分解公式，进一步引入了D关于其Metzler矩阵a (D)的l函数，并给出了它的行列式表达式。作为推论，我们给出了由l函数覆盖H (D)的群的Metzler矩阵a (H)的行列式的分解公式。

引用次数: 0

Quasi-cyclic codes of index 2 索引2的拟循环码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-19 DOI: 10.1016/j.disc.2026.115004

Kanat Abdukhalikov , Askar S. Dzhumadil'daev , San Ling

We study quasi-cyclic codes of index 2 over finite fields. We give a classification of such codes. Their duals with respect to the Euclidean, symplectic and Hermitian inner products are investigated. We describe self-orthogonal and dual-containing codes. Lower bounds for minimum distances of quasi-cyclic codes are given. A quasi-cyclic code of index 2 is generated by at most two elements. We describe conditions when such a code (or its dual) is generated by one element.

研究有限域上指标2的拟循环码。我们对这类代码进行了分类。研究了它们对欧几里得内积、辛内积和厄米内积的对偶。我们描述了自正交和双包含码。给出了拟循环码最小距离的下界。索引为2的拟循环码由最多两个元素生成。我们将描述由一个元素生成这种代码（或其对偶）的条件。

引用次数: 0

On DP-coloring of outerplanar graphs 关于外平面图的dp染色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-16 DOI: 10.1016/j.disc.2026.114996

Tianjiao Dai , Jie Hu , Hao Li , Shun-ichi Maezawa

The notion of DP-coloring was introduced by Dvořák and Postle which is a generalization of list coloring. A DP-coloring of a graph G reduces the problem of finding a proper coloring of G from a given list L to the problem of finding a “large” independent set in an auxiliary graph

M_{L}

-cover with a vertex set

{(v, c) : v \in V (G)

and

c \in L (v)}

. Hutchinson (Journal of Graph Theory, 2008) showed that

•
if a 2-connected bipartite outerplanar graph G has a list of colors $L (v)$ for each vertex v with $| L (v) | \geq \min {\deg_{G} (v), 4}$ , then G is L-colorable; and
•
if a 2-connected maximal outerplanar graph G with at least four vertices has a list of colors $L (v)$ for each vertex v with $| L (v) | \geq \min {\deg_{G} (v), 5}$ , then G is L-colorable.

In this paper, we study whether bounds of Hutchinson's results hold for DP-coloring. We obtain that the first one is not sufficient for DP-coloring while the second one is sufficient.

dp -着色的概念是由Dvořák和Postle提出的，它是列表着色的推广。图G的dp -着色将从给定列表L中寻找G的适当着色问题简化为在具有顶点集{(v,c):v∈v (G) and c∈L(v)}的辅助图ML-cover中寻找“大”独立集的问题。Hutchinson （Journal of Graph Theory, 2008）证明了•如果一个2连通二部外平面图G对于每个顶点v有一个颜色列表L(v)且|L(v)|≥min (degG) {degG (v),4}，则G是L可色的；•如果一个至少有四个顶点的2连通最大外平面图G对每个顶点v都有一个颜色列表L(v)，且|L(v)|≥min (degG) (v),5}，则G是L可色的。在本文中，我们研究了Hutchinson结果的界对于dp -着色是否成立。我们得到第一个是不充分的，而第二个是充分的。

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