Discrete Applied Mathematics最新文献

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On edge colorings of graphs with no color-rich cycles

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.030

Tomáš Madaras, Alfréd Onderko

Motivated by the broad studies of various edge-coloring anti-Ramsey-like problems where particular subgraphs are forbidden to be colored in a rainbow way or using many colors in general, we investigate edge-colorings with the constraint that each cycle sees at most

i

colors, for a given positive integer

i

. For such a coloring, the goal is to find the maximum number of colors, denoted by

K_{i}^{\circ} (G)

. We explore the basic properties and estimates of

K_{i}^{\circ} (G)

. A greedy-based general lower bound on this invariant is provided and its sharpness is discussed; we show that, for small values of

i

, this bound is attained for highly connected graphs.

引用次数: 0

The quickest root–leaf interdiction problem on tree networks

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.020

Huong Nguyen-Thu , Javad Tayyebi , Kien Trung Nguyen , Nguyen Thanh Luan

We investigate the quickest root–leaf interdiction problem on a tree, in which two players compete to achieve their objectives. The evader aims to escape the tree by traversing from the root to one of the leaves with the minimum transmission time, while the interdictor seeks to reduce the edge capacities within a given budget to maximize the transmission time of the evader in the perturbed tree. We prove that if the modifying cost is measured using the Hamming distance and the adjusted capacities are limited to a threshold, then there is an optimal solution, where the modified capacities are either equal to themselves or equal to the threshold. We also demonstrate that there are a linearly large number of values that can serve as candidates for the optimal objective. For each fixed value, we construct an auxiliary network in which its minimum cut is associated with the given value. The minimum cut approach helps to develop a combinatorial algorithm that solves the corresponding problem in

O (n max {m, log n})

time, where

n

and

m

are respectively the number of vertices and leaves in the underlying tree.

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引用次数: 0

Maximal first Zagreb connection index of trees with given total domination number

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.031

Zahid Raza , Shehnaz Akhter , Roslan Hasni

Chemical graph theory plays a crucial role in mathematical chemistry by using graph invariants to represent chemical phenomena in a mathematical framework. Topological descriptors, which are graph invariants formed from molecular graph representations of chemical compounds, are employed in QSPR and QSAR studies. The Zagreb connection indices are topological descriptors which are used to analyze graphs based on connection cardinality. They were introduced in 1972 to calculate the total electron energy of alternate hydrocarbons. Recently, they have gained renewed attention due to their ability to provide a more precise correlation for the physicochemical characteristics of different compounds compared to basic Zagreb indices. In this article, we aim to analyze and establish the correlations of the first Zagreb connection index and the total domination number of trees. Our focus is on providing a detailed description of trees having the largest first Zagreb connection index among trees that have a specific total domination number.

引用次数: 0

The minimum number of maximal independent sets in graphs with given order and independence number

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.027

Yuting Tian, Jianhua Tu

Let

M I S (G)

be the set of all maximal independent sets in a graph

G

, and let

m i s (G) = | M I S (G) |

. In this paper, we show that for any tree

T

with

n

vertices and independence number

α

m i s (T) \geq f (n - α + 2),

and for any unicyclic graph

G

with

n

vertices and independence number

α

m i s (G) \geq (\begin{matrix} 2, & if n = 4, \\ 3, & if α = n - 2 and n \neq 4, \\ 2 f (n - α), & if n \geq 6 and ⌈ \frac{n}{2} ⌉ \leq α < n - 2, \\ f (n - α + 2) - f (n - α - 3), & if n \geq 5, and n is odd, and α = ⌊ \frac{n}{2} ⌋, \end{matrix})

where

f (n)

represents the

n

th Fibonacci number. Moreover, we also show that the above inequalities are sharp.

引用次数: 0

On prescribed Hamilton laceability of hybrid-faulty star graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.018

Zai Ping Lu, Shu Dan Xue

<div><div>In this paper, we investigate Hamilton paths passing through a prescribed linear forest in a hybrid-faulty star graph. Let <math><mi>M</mi></math> be a matching with <math><mi>m</mi></math> edges of the star graph <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, <math><mi>F</mi></math> be an <math><mi>f</mi></math>-subset of <math><mrow><mi>E</mi><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math>, <math><mrow><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>}</mo></mrow></math> be a 2-subset of <math><mrow><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math>, and <math><mi>L</mi></math> be a linear forest of <math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>−</mo><mi>F</mi></mrow></math>. Suppose that <math><mrow><mi>m</mi><mo>+</mo><mi>f</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>4</mn></mrow></math>, neither <math><mi>x</mi></math> nor <math><mi>y</mi></math> is an inner vertex of <math><mi>L</mi></math>, and <math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math> are not located on the same component of <math><mi>L</mi></math>. We prove that, for any <math><mrow><mi>w</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>∖</mo><mi>V</mi><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow></mrow></math>, there exists a Hamilton path of <math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>−</mo><mi>F</mi><mo>−</mo><mi>w</mi></mrow></math> between <math><mi>x</mi></math> and <math><mi>y</mi></math> passing through <math><mi>L</mi></math> provided that <math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math> belong to the partite set not containing <math><mi>w</mi></math>, and <math><mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>−</mo><mi>m</mi><mo>−</mo><mi>f</mi></mrow></math>. As a consequence, if <math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math> are from opposite partite sets and <math><mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>−</mo><mi>m</mi><mo>−</mo><mi>f</mi></mrow></math></

引用次数: 0

On the max min of the algebraic degree and the nonlinearity of a Boolean function on an affine subspace

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.022

Jan Kristian Haugland

We investigate the max min of the algebraic degree and the nonlinearity of a Boolean function in

n

variables when restricted to a

k

-dimensional affine subspace of

F_{2}^{n}

. Previous authors have focused on the cases when the max min of the algebraic degree is 0 or 1. Upper bounds, lower bounds and a conjecture on the exact value in special cases are presented.

引用次数: 0

A note on the online hierarchical scheduling for lp-norm load balancing

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-27 DOI: 10.1016/j.dam.2025.02.029

Xianglai Qi, Lingfa Lu, Shiguo Zhou

In this paper, we consider the online hierarchical scheduling for load balancing on two identical machines. In the problem, the jobs are available online over list and the objective is to minimize the

l_{p}

-norm of every machine’s load. A best possible online algorithm is designed for this problem.

引用次数: 0

General degree-eccentricity index of unicyclic graphs of given order, girth and maximum degree

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-21 DOI: 10.1016/j.dam.2025.02.017

Carl Johan Casselgren , Mesfin Masre

For a connected graph

G

and

a, b \in R

, the general degree-eccentricity index of

G

is defined as

D E I_{a, b} (G) = \sum_{v \in V (G)} d_{G}^{a} (v) e c c_{G}^{b} (v)

, where

V (G)

is the vertex set of

G

d_{G} (v)

is the degree of a vertex

v

and

e c c_{G} (v)

is the eccentricity of

v

G

, i.e. the maximum distance from

v

to another vertex of the graph. This index generalizes several well-known ‘topological indices’ of graphs such as the eccentric connectivity index. We characterize the unique unicyclic graphs with the maximum and the minimum general degree-eccentricity index among all

n

-vertex unicyclic graphs with fixed order, girth, and maximum degree for the cases

a \geq 1, b \leq 0

and

0 \leq a \leq 1, b \geq 0

引用次数: 0

An optimal chromatic bound for the class of {P3∪2K1,P3∪2K1¯}-free graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-21 DOI: 10.1016/j.dam.2025.02.006

Athmakoori Prashant , S. Francis Raj

In 1987, A. Gyárfás in his paper “Problems from the world surrounding perfect graphs” posed the problem of determining the smallest

χ

-binding function for

G (F, \bar{F})

, when

G (F)

χ

-bounded. So far the problem has only been attempted for some forest

F

with four or five vertices. In this paper, we address the problem when

F = P_{3} \cup 2 K_{1}

and show that if

G

is a

{P_{3} \cup 2 K_{1}, \bar{P_{3} \cup 2 K_{1}}}

-free graph with

ω (G) \neq 3

, then it admits

ω (G) + 1

as a

χ

-binding function. Moreover, we also construct examples to show that this bound is tight for all values of

ω \neq 3

引用次数: 0

A synthesis for uniformly 3-edge-connected graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-02-20 DOI: 10.1016/j.dam.2025.02.005

Carl Kingsford , Guillaume Marçais

A multigraph is uniformly

k

-edge-connected if there are exactly

k

edge-disjoint paths between any pair of vertices. For example, a uniformly

k

-edge-connected graph is obtained from a

k

-edge-connected graph by collapsing the nodes connected by more than

k

edge-disjoint paths into supernodes. We characterize the class of uniformly 3-edge-connected graphs, giving a synthesis involving two operations by which every uniformly 3-edge-connected multigraph can be generated. Slightly modified syntheses give the planar uniformly 3-edge-connected graphs and the uniformly 3-edge-connected graphs with the fewest possible edges, generalizing the well-known Harary graphs. In proving the correctness of the synthesis, we also show the existence of a particular type of induced, non-separating cycle in near 3-regular graphs, which is of interest in its own right.

引用次数: 0

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