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Solution to a 3-path isolation problem for subcubic graphs 次三次图的三路径隔离问题的解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.disc.2025.114970

Karl Bartolo, Peter Borg, Dayle Scicluna

The 3-path isolation number of a connected n-vertex graph G, denoted by

ι (G, P_{3})

, is the size of a smallest subset D of the vertex set of G such that the closed neighbourhood

N [D]

of D intersects the vertex sets of the 3-vertex paths of G, meaning that no two edges of

G - N [D]

intersect. If G is not a 3-path or a 3-cycle or a 6-cycle, then

ι (G, P_{3}) \leq 2 n / 7

. This was proved by Zhang and Wu, and independently by Borg in a slightly extended form. The bound is attained by infinitely many connected graphs having induced 6-cycles. Huang, Zhang and Jin showed that if G has no 6-cycles, or G has no induced 5-cycles and no induced 6-cycles, then

ι (G, P_{3}) \leq n / 4

unless G is a 3-path or a 3-cycle or a 7-cycle or an 11-cycle. They asked if the bound still holds asymptotically for connected graphs having no induced 6-cycles. Thus, the problem essentially is whether induced 6-cycles solely account for the difference between the two bounds. In this paper, we solve this problem for subcubic graphs, which need to be treated differently from other graphs. We show that if G is subcubic and has no induced 6-cycles, then

ι (G, P_{3}) \leq n / 4

unless G is a copy of one of 12 particular graphs whose orders are 3, 7, 11 and 15. The bound is sharp.

连通的N顶点图G的3路隔离数，用ι(G,P3)表示，是G的顶点集的最小子集D的大小，使得D的闭邻域N[D]与G的3顶点路径的顶点集相交，即G−N[D]没有两条边相交。若G不是3径、3环或6环，则ι(G,P3)≤2n/7。这是由Zhang和Wu证明的，Borg以一个稍微扩展的形式独立地证明了。该界是由无穷多个具有诱导6环的连通图得到的。Huang， Zhang和Jin证明了如果G没有6环，或者G没有诱导5环和诱导6环，那么除非G是3径或3环或7环或11环，否则ι(G,P3)≤n/4。他们问，对于没有诱导6环的连通图，界是否仍然是渐近成立的。因此，问题本质上是诱导的6环是否完全解释了两个界之间的差异。在本文中，我们解决了亚三次图的这个问题，它需要区别于其他图的处理。我们证明了如果G是次立方的并且没有诱导的6环，那么ι(G,P3)≤n/4，除非G是阶数为3,7,11和15的12个特定图之一的副本。边界是尖锐的。

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引用次数: 0

Constructions of minimally t-tough regular graphs 最小t-坚韧正则图的构造

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.disc.2026.114993

Kun Cheng , Chengli Li , Feng Liu

A non-complete graph G is said to be t-tough if for every vertex cut S of G, the ratio of

| S |

to the number of components of

G - S

is at least t. A complete graph is said to be t-tough for any

t > 0

. The toughness

τ (G)

of the graph G is the maximum value of t such that G is t-tough. A graph G is said to be minimally t-tough if

τ (G) = t

and

τ (G - e) < t

for every

e \in E (G)

. In 2003, Kriesell conjectured that every minimally 1-tough graph contains a vertex of degree 2. In 2018, Katona and Varga generalized this conjecture, asserting that every minimally t-tough graph contains a vertex of degree

⌈ 2 t ⌉

. Recently, Zheng and Sun disproved the generalized Kriesell conjecture by constructing a family of 4-regular graphs of even order. They also raised the question of whether there exist other minimally t-tough regular graphs that do not satisfy the generalized Kriesell conjecture. In this paper, we provide an affirmative answer by constructing a family of 4-regular graphs of odd order, as well as a family of 6-regular graphs of order

3 k + 1

, where

k \geq 5

如果对于G的每个顶点切割S， |S|与G−S的分量数之比至少为t，则称非完全图G是t-tough的。对于任意t>；0，称完全图G是t-tough的。图G的韧性τ(G)是t的最大值，使得G为t-tough。如果τ(G)=t且对于每一个e∈e (G) τ(G−e)<t，则图G是最小t-坚韧的。2003年，Kriesell推测每个最小1-tough图都包含一个2次顶点。2018年，Katona和Varga推广了这一猜想，断言每个最小t-tough图都包含一个度为≤2t²的顶点。最近，郑和孙通过构造一组偶阶的4正则图来否定广义Kriesell猜想。他们还提出了是否存在其他不满足广义Kriesell猜想的最小t-坚韧正则图的问题。本文通过构造一个奇阶4正则图族，以及k≥5阶3k+1阶6正则图族，给出了一个肯定的答案。

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引用次数: 0

On tetravalent 3-geodesic transitive graphs 关于四价3测地线传递图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.disc.2026.114995

Jun-Jie Huang , Yan-Quan Feng , Jin-Xin Zhou

For

s \geq 1

and a graph Γ, a sequence

(u_{0}, u_{1}, \dots, u_{s})

of vertices of Γ is called an s-arc if

u_{i}

is adjacent to

u_{i + 1}

for

0 \leq i \leq s - 1

and

u_{i - 1} \neq u_{i + 1}

for

1 \leq i \leq s - 1

, and an s-geodesic if in addition

u_{0}

and

u_{s}

has distance s. We say that Γ is s-arc transitive if its automorphism group is transitive on the set of s-arcs, and s-geodesic transitive if for each

i \leq s

, Γ has at least one i-geodesic, and its automorphism group is transitive on the set of i-geodesics for all

i \leq s

. In Jin (2015) [15], Jin initiated the investigation of tetravalent 3-geodesic transitive but not 3-arc transitive graphs, and he classified all such graphs of girth at most 4 and conjectured that there do not exist such graphs of girth 5. In this paper, a complete classification is given of tetravalent 3-geodesic transitive but not 3-arc transitive graphs. As a result, we find that all such graphs have girth at most 5, and that there exist four such graphs of girth 5. This disproves Jin's conjecture.

s≥1和图Γ序列(美国情况,u1,⋯)的顶点Γ称为s-arc如果ui毗邻ui + 1 0≤≤s−1和ui−1≠ui + 1对1≤≤s−1,和一个s-geodesic如果此外情况和我们已经距离。我们说Γs-arc传递如果其自同构群是s-arcs,传递和s-geodesic传递如果每个我≤年代,Γ至少有一个i-geodesic,和其自同构群传递的i-geodesics我≤年代。Jin(2015)[15]中，Jin率先研究了四价3-测地线传递图而非3-弧传递图，他将所有最多4周长的图分类，并推测不存在5周长的图。本文给出了四价3测地线传递图的完全分类，但没有给出3弧传递图的完全分类。结果，我们发现所有这样的图的周长最多为5，并且存在4个这样的图的周长为5。这推翻了金的猜想。

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引用次数: 0

On non-traceable 3-connected planar cubic graphs of minimum order 最小阶不可追溯三连通平面三次图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-12 DOI: 10.1016/j.disc.2026.114983

Gholamreza Abrishami , Freydoon Rahbarnia , Nico Van Cleemput

In 1980, Zamfirescu presented a non-traceable (i.e. without a hamiltonian path) 3-connected planar cubic graph with 88 vertices, which is still the smallest known one of its kind. In this note we present several new examples with 88 vertices. Moreover, we present a non-traceable cyclically 4-connected planar cubic graph with 168 vertices.

1980年，Zamfirescu提出了一个88个顶点的不可追踪（即没有哈密顿路径）三连通平面三次图，这是目前已知的最小的三连通平面三次图。在这篇文章中，我们提供了几个新的88个顶点的例子。此外，我们给出了一个不可追踪的有168个顶点的环四连通平面三次图。

引用次数: 0

Strictly k-colorable graphs 严格k色图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-09 DOI: 10.1016/j.disc.2026.114980

Evan Leonard

Zhu [5] introduced a refined scale of choosability in 2020 and observed that the four color theorem is tight on this scale. We formalize and explore this idea of tightness in what we call strictly colorable graphs. We then characterize all strictly colorable complete multipartite graphs.

Zhu[5]在2020年引入了一个精细的可选择性尺度，并观察到四色定理在这个尺度上是紧密的。我们在所谓的严格可色图中形式化并探索紧性的概念。然后我们刻画了所有严格可着色完全多部图。

引用次数: 0

Structure and linear-Pollyanna for some square-free graphs 一些无平方图的结构和线性波利安娜

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.disc.2026.114979

Ran Chen, Baogang Xu

We use

P_{t}

and

C_{t}

to denote a path and a cycle on t vertices, respectively. A bull is a graph consisting of a triangle with two disjoint pendant edges, a hammer is a graph obtained by identifying an endvertex of a

P_{3}

with a vertex of a triangle. A class

F

is χ-bounded if there is a function f such that

χ (G) \leq f (ω (G))

for all induced subgraphs G of a graph in

F

. A class

C

of graphs is Pollyanna (resp. linear-Pollyanna) if

C \cap F

is polynomially (resp. linearly) χ-bounded for every χ-bounded class

F

of graphs. Chudnovsky et al. [6] showed that both the classes of bull-free graphs and hammer-free graphs are Pollyannas. Let G be a connected graph with no clique cutsets and no universal vertices. In this paper, we show that G is

(C_{4}

, hammer)-free if and only if it has girth at least 5, and G is

(C_{4}

, bull)-free if and only if it is a clique blowup of some graph of girth at least 5. As a consequence, we show that both the classes of

(C_{4}

, bull)-free graphs and

(C_{4}

, hammer)-free graphs are linear-Pollyannas. We also show that the class of (bull, diamond)-free graphs is linear-Pollyanna.

我们用Pt和Ct分别表示t个顶点上的路径和循环。牛是由一个三角形的两个不相交的垂边组成的图，锤是由P3的一个端点与一个三角形的一个顶点确定而成的图。如果存在一个函数F，使得F中一个图的所有诱导子图G的χ(G)≤F (ω(G))，则该类F是χ-有界的。如果C∩F是多项式的(p。对于每一个有χ有界的图类F。Chudnovsky et al.[6]表明无牛图和无锤图都是盲目乐观的。设G是一个连通图，没有团切集，也没有全称顶点。本文证明了G是（C4，锤子）自由的当且仅当它的周长至少为5，并且G是（C4，公牛）自由的当且仅当它是某个周长至少为5的图的团团爆破。因此，我们证明了（C4，牛）自由图和（C4，锤）自由图都是线性的盲目乐观。我们还证明了一类（牛，菱形）无图是线性的。

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引用次数: 0

Minimal common factor graphs containing all graphs of order k 包含所有k阶图的最小公因式图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.disc.2025.114974

G. Batta , L. Hajdu

We are concerned with the minimal representation of graphs as common factor graphs. First we show that for any graph G of order k one can find

a_{1}, a_{2}, \dots, a_{k} \in N

to represent G such that the number of prime divisors of

a_{1} a_{2} \dots a_{k}

is at most

⌊ k^{2} / 4 ⌋

, and that this value is best possible. Then we give upper and lower bounds (which differ only in a constant factor in the exponent) for the smallest n such that every graph of order k is an induced subgraph of the common factor graph induced by the set

{1, 2, \dots, n}

. Further, we answer a question of Eggleton from 1987 concerning graphs which are extremal for this type of representability to the negative, formulate a conjecture containing three assertions, and provide some related numerical results.

我们关注图作为公因子图的最小表示。首先，我们证明了对于任何k阶图G，我们可以找到a1，a2，…，ak∈N来表示G，使得a1a2，ak的素数最多为⌊k2/4⌋，并且这个值是最佳可能值。然后，我们给出最小n的上界和下界（仅在指数上有一个常数因子的区别），使得每个k阶图都是由集合{1,2，…，n}诱导的公因子图的诱导子图。进一步，我们回答了Eggleton在1987年提出的关于这种类型的负可表征性的极值图的问题，提出了一个包含三个断言的猜想，并提供了一些相关的数值结果。

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引用次数: 0

Fixed perimeter analogues of some partition results 一些分区结果的固定周长类似物

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.disc.2025.114968

Gabriel Gray , Emily Payne , Holly Swisher , Ren Watson

Euler's partition identity states that the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook length (perimeter) n. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.

欧拉划分恒等式指出n被划分为奇数部分的个数等于n被划分为不同部分的个数。引人注目的是，Straub在2016年证明，当计算具有最大钩子长度（周长）n的任何大小的分区时，这个恒等式也成立。这激发了对固定周长设置下的分区恒等式和不等式的进一步研究。在这里，我们探索一些著名的分割结果的固定周长类似于欧拉的分割恒等式。

引用次数: 0

Solutions to open problems on the exponential augmented Zagreb index 指数增广萨格勒布指数开放性问题的解法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.disc.2025.114967

Kinkar Chandra Das , Da-yeon Huh , Sourav Mondal

The exponential augmented Zagreb (EAZ) index is a graph-theoretical descriptor that correlates strongly with the physico-chemical properties of molecules. Introduced by Rada in 2019, it is defined for a simple graph ϒ as

E A Z (ϒ) = \sum_{v_{i} v_{j} \in E (ϒ)} e^{{(\frac{d_{i} d_{j}}{d_{i} + d_{j} - 2})}^{3}},

where

E (ϒ)

denotes the edge set and

d_{i}

is the degree of vertex

v_{i}

. This work is motivated by some open problems concerning the well-known augmented Zagreb index (AZ). In particular, the maximization of AZ for a given graph order and a specified number of pendant vertices was posed as an open problem in Chen et al. (2022) [7]. We completely resolve this problem for the exponential version, EAZ. In recent work Xu et al. (2025) [38], two related questions were raised: whether the maximal graphs for AZ and EAZ coincide, and if not, how they differ. We provide complete answers to these questions with respect to the chromatic number and the number of pendant vertices. We explore the maximal graph for EAZ in terms of chromatic number and graph order, and show that this differs substantially from the corresponding extremal graph for AZ. Further results include a characterization of the maximal graphs for EAZ when vertex connectivity and edge connectivity are prescribed together with the graph order. In addition, we prove that

E A Z (ϒ)

increases upon adding an edge to ϒ, a crucial result for understanding the extremal properties of EAZ. Finally, the potential usefulness of this discrete invariant in chemical graph theory is demonstrated.

指数增强萨格勒布指数（EAZ）是一种与分子的物理化学性质密切相关的图理论描述符。由Rada于2019年引入，它被定义为一个简单的图表γ asEAZ(y)=∑vivj∈E(y) E(didjdi+dj−2)3，其中E(y)表示边缘集，di表示顶点vi的度。这项工作的灵感来自于一些关于著名的增强萨格勒布指数（AZ）的开放问题。特别是，在Chen等人（2022）[7]中，给定图阶和指定数量的垂顶点的AZ最大化是一个开放问题。我们完全解决了这个问题的指数版本，EAZ。在最近的工作Xu et al. (2025) b[38]中，提出了两个相关的问题：AZ和EAZ的最大图是否重合，如果不重合，它们是如何不同的。我们提供了关于色数和垂顶点数的完整答案。我们从色数和图阶的角度探讨了EAZ的极大图，并表明这与AZ的相应极值图有很大的不同。进一步的结果包括了当顶点连通性和边连通性与图阶一起规定时EAZ的极大图的表征。此外，我们证明了在给y增加一个边缘后，y的值会增加，这是理解y的极端属性的一个关键结果。最后，证明了该离散不变量在化学图论中的潜在用途。

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引用次数: 0

Spectral extrema of graphs with fixed size: Forbidden star forests 固定尺寸图的谱极值：禁星森林

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.disc.2025.114976

Yanting Zhang , Ligong Wang

The spectral radius of a graph G, denoted by

ρ (G)

, is the largest eigenvalue of its adjacency matrix. The Brualdi-Hoffman-Turán type problem is to determine the maximum spectral radius among all m-edge graphs which do not contain specific forbidden subgraphs. Denote by

S_{ℓ}

the star on

ℓ + 1

vertices. Let F be a star forest, where

F = \cup_{i = 1}^{k} S_{ℓ_{i}}

with

k \geq 2

and

ℓ_{i} \geq 1

for

i \in [k]

. In this paper, we study the Brualdi-Hoffman-Turán type problem for star forests, and prove that if G is an F-free graph with size m, then its spectral radius satisfies

ρ (G) \leq \frac{1}{2} (k - 2 + \sqrt{4 m - k^{2} + 2 k})

, with equality if and only if

G = K_{k - 1} \lor (\frac{m}{k - 1} - \frac{k - 2}{2}) K_{1}

, provided that

m \geq {(2 k - 1)}^{2} {(\sum_{i = 1}^{k} ℓ_{i} + k - 2)}^{2}

图G的谱半径，用ρ(G)表示，是其邻接矩阵的最大特征值。Brualdi-Hoffman-Turán类型问题是确定所有不包含特定禁止子图的m边图的最大谱半径。用S表示在1个顶点上的星号。设F是一个星林，其中对于i∈[k]， F=∪i= 1ks_i，且k≥2且_i≥1。本文研究了星林的Brualdi-Hoffman-Turán型问题，证明了若G是大小为m的无f图，则其谱半径满足ρ(G)≤12(k−2+4m−k2+2k)，且当且仅当G=Kk−1∨(mk−1−k−22)K1，且m≥(2k−1)2(∑i=1k∑i +k−2)2。

引用次数: 0

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