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9 pages, 464 KB

Open AccessArticle

Hadwiger’s Conjecture for Dense Strongly Regular Graphs

by Guangjun Xu, Lijuan Lei, Xianghu Liu and Yanfang Li

Symmetry 2025, 17(10), 1588; https://doi.org/10.3390/sym17101588 - 23 Sep 2025

Viewed by 1265

The famous Four-Color Conjecture (now Theorem) states that any planar graph could be colored using four colors. Hadwiger’s conjecture strengthens the Four-Color Conjecture by asserting that every graph with chromatic number t contains a complete minor of order t. In this paper we investigate Hadwiger’s conjecture for the complements of the Petersen graph and the Clebsch graph; both are strongly regular graphs with independence number two (hence dense graphs). We confirm Hajós’ conjecture, hence Hadwiger’s conjecture, for these graphs. Moreover, we show that for each of these graphs the exact hadwiger number is strictly greater its chromatic number. Full article

(This article belongs to the Special Issue Symmetry in Graph Algorithms and Graph Theory III)

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14 pages, 255 KB

Open AccessArticle

A Characterization of Three-Dimensional Convex Polytopes with Five Pairwise Antipodal Vertices

by Rong Guo, Chan He, Longzhen Zhang and Senlin Wu

Mathematics 2025, 13(9), 1412; https://doi.org/10.3390/math13091412 - 25 Apr 2025

Viewed by 957

Abstract

Concerning the antipodality properties of finite sets, we focus on convex polytopes in

R^{3}

with less than

2^{3}

vertices and characterize convex polytopes with 5 vertices that are pairwise antipodal. Full article

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12 pages, 279 KB

Open AccessArticle

Construction of ε-Nets for the Space of Planar Convex Bodies Endowed with the Banach–Mazur Metric

by Yanmei Chen, Yunfang Lyu, Shenghua Gao and Senlin Wu

Mathematics 2025, 13(8), 1358; https://doi.org/10.3390/math13081358 - 21 Apr 2025

Viewed by 811

Abstract

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et [...] Read more.

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of

ε

-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing

ε

-nets for

(K^{n}, d_{B M})

based on finite subsets of

Z^{n}

theoretically. In this work, we present an algorithm to construct

ε

-nets for

(K^{2}, d_{B M})

and a

(1 / 4)

-net for

(C^{2}, d_{B M})

is constructed. To the best of our knowledge, this is the first concrete

ε

-net for

(C^{2}, d_{B M})

for such a small

ε

. Full article

(This article belongs to the Section B: Geometry and Topology)

23 pages, 360 KB

Open AccessArticle

Homothetic Covering of Crosspolytopes

by Yunfang Lyu, Feifei Chen and Senlin Wu

Mathematics 2025, 13(4), 546; https://doi.org/10.3390/math13040546 - 7 Feb 2025

Cited by 1 | Viewed by 889

Abstract

The exact value of

Γ_{m} (K)

, which is the least positive number

γ

such that a convex body K can be covered by m translates of

γ K

, is usually difficult to obtain. We present exact values of [...] Read more.

The exact value of

Γ_{m} (K)

, which is the least positive number

γ

such that a convex body K can be covered by m translates of

γ K

, is usually difficult to obtain. We present exact values of

Γ_{14} (B_{1}^{3})

Γ_{11} (B_{1}^{4})

Γ_{2 n} (B_{1}^{n})

Γ_{2 n + 1} (B_{1}^{n})

, and

Γ_{2 n + 2} (B_{1}^{n})

, where

B_{1}^{n}

is the unit ball of

R^{n}

endowed with the taxicab norm. Full article

(This article belongs to the Section B: Geometry and Topology)

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16 pages, 318 KB

Open AccessArticle

An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies

by Xiangyang Han, Senlin Wu and Longzhen Zhang

Axioms 2024, 13(2), 132; https://doi.org/10.3390/axioms13020132 - 19 Feb 2024

Cited by 3 | Viewed by 1851

Abstract

In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron. Full article

(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)

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15 pages, 388 KB

Open AccessArticle

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

by Man Yu, Yafang Lv, Yanping Zhao, Chan He and Senlin Wu

Mathematics 2023, 11(9), 2000; https://doi.org/10.3390/math11092000 - 23 Apr 2023

Cited by 5 | Viewed by 1908

Abstract

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented. Full article

(This article belongs to the Section B: Geometry and Topology)

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