On the neighbor full sum distinguishing total coloring of graphs

Fast algorithm for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e44" altimg="si15.svg"> <mml:mi>S</mml:mi> </mml:math> -packing coloring of Halin graphs

On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e85" altimg="si17.svg"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -packing coloring of some subcubic graphs

The labor- and experience-intensive creation of 3D assets with physically based rendering (PBR) materials demands an autonomous 3D asset creation pipeline. However, most existing 3D generation methods focus on geometry modeling, either baking textures into simple vertex colors or leaving texture synthesis to post-processing with image diffusion models. To achieve end-to-end PBR-ready 3D asset generation, we present Lightweight Gaussian Asset Adapter (LGAA), a novel framework that unifies the modeling of geometry and PBR materials by exploiting multi-view (MV) diffusion priors from a novel perspective. The LGAA features a modular design with three components. Specifically, the LGAA Wrapper reuses and adapts network layers from MV diffusion models, which encapsulate knowledge acquired from billions of images, enabling better convergence in a data-efficient manner. To incorporate multiple diffusion priors for geometry and PBR synthesis, the LGAA Switcher aligns multiple LGAA Wrapper layers encapsulating different knowledge. Then, a tamed variational autoencoder (VAE), termed LGAA Decoder, is designed to predict 2D Gaussian Splatting (2DGS) with PBR channels. Finally, we introduce a dedicated post-processing procedure to effectively extract high-quality, relightable mesh assets from the resulting 2DGS. Extensive quantitative and qualitative experiments demonstrate the superior performance of LGAA with both text- and image-conditioned MV diffusion models. Additionally, the modular design enables flexible incorporation of multiple diffusion priors, and the knowledge-preserving scheme effectively preseves the 2D priors learned on massive image dataset, which leads to data efficient finetuning to lift the MV diffuison models for 3D generation with merely 69 k multi-view instances.

&lt;p&gt;An equitable vertex &lt;span class="math inline"&gt;\(k\)&lt;/span&gt;-coloring of a graph &lt;span class="math inline"&gt;\(G\)&lt;/span&gt; is a proper vertex coloring with &lt;span class="math inline"&gt;\(k\)&lt;/span&gt; colors such that the size of any two color classes differ by at most one. It was conjectured by Meyer in 1973 that every graph other than the complete graph &lt;span class="math inline"&gt;\(K_n\)&lt;/span&gt; or an odd cycle &lt;span class="math inline"&gt;\(C_{2m+1}\)&lt;/span&gt; admits an equitable vertex coloring with at most &lt;span class="math inline"&gt;\(\Delta(G)\)&lt;/span&gt; colors, where &lt;span class="math inline"&gt;\(\Delta(G)\)&lt;/span&gt; is the maximum degree of a vertex in graph &lt;span class="math inline"&gt;\(G\)&lt;/span&gt;. We show that the conjecture is true for middle graphs of some cycle-based graphs.&lt;/p&gt;

A proper coloring ϕ of G is called a proper conflict-free coloring of G if for every non-isolated vertex v of G , there is a color c such that | ϕ − 1 ( c ) ∩ N G ( v ) | = 1 . As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free ( degree + k ) -choosability of graphs. For a non-negative integer k , a graph G is proper conflict-free ( degree + k ) -choosable if for any list assignment L of G with | L ( v ) | ≥ d G ( v ) + k for every vertex v ∈ V ( G ) , G admits a proper conflict-free coloring ϕ such that ϕ ( v ) ∈ L ( v ) for every vertex v ∈ V ( G ) . In this note, we first remark if a graph G is d -degenerate, then G is proper conflict-free ( degree + d + 1 ) -choosable. Furthermore, when d = 1 , we can reduce the number of colors by showing that every tree is proper conflict-free ( degree + 1 ) -choosable. This motivates us to state a question.

A proper k -coloring of a graph is said to be odd if every non-isolated vertex has a color that appears an odd number of times on its neighborhood. Miao et al. (2024) [2] claimed that every planar graph without adjacent 3-cycles is odd 7-colorable and every triangle-free planar graph without intersecting 4-cycles is odd 5-colorable. Here, we point out that their published proof contains a fundamental flaw which affects the validity of the main results.

It is known that, for an oriented hypergraph with (vertex) coloring number χ and smallest and largest normalized Laplacian eigenvalues λ₁ and λ_N, respectively, the inequality χ≥(λ_N−λ₁)/min⁡{λ_N−1,1−λ₁} holds. We provide necessary conditions for oriented hypergraphs for which this bound is tight. Focusing on c-uniform unoriented hypergraphs, we then generalize the bound to the setting of d-proper colorings: colorings in which no edge contains more than d vertices of the same color. We also adapt our proof techniques to derive analogous spectral bounds for d-improper colorings of graphs and for edge colorings of hypergraphs. Moreover, for all coloring notions considered, we provide necessary conditions under which the bound is an equality.

Tree graph is a connected graph and has no circuits. Tree graphs used in this study include: broom graph, firework graph, banana tree graph, centipede graph, E graph, and double star graph. Graph coloring is the process of giving color to graph elements with the rule that neighboring elements must not have the same color and the number of colors used must be as minimal as possible. b-Coloring of a graph G is a coloring of the vertices of G such that each color class has at least one vertex adjacent to all other color classes. The b-Chromatic number of a graph G is denoted by φ (G), is the largest integer k such that G has a b-coloring with k colors. The limit of b-coloring of graph G with maximum degree ∆(G) is as follows. χ(G)≤φ(G)≤∆(G)+1.χ(G) is the chromatic number of a graph G where χ(G) is the minimum value of the color required for proper coloring of graph G. While ∆(G) is the maximum degree of the vertices in graph G. This study uses an exploratory research type with axiomatic deductive method and pattern detection method. Based on the results of this study, the results of the b-coloring analysis on the tree graph family are known. The results of this study are expected to be used as study material and development of scientific knowledge related to b-coloring of other graphs

The sustainability of wireless sensor networks critically depends on intelligent and efficient charger deployment. This paper proposes a two-stage optimization framework that determine the total number of chargers needed to recharge and the optimal position of charger. In the first stage, a hybrid algorithm combining degree-based saturation and Grundy coloring efficiently determines the minimal number of chargers required. In the second stage, an Enhanced Aquila Optimization algorithm, inspired by eagle hunting behavior, identifies optimal charger locations under coverage and power constraints. Experimental results show that the enhancement method improves coverage by 6% compared to the standard algorithm. The Enhanced Aquila Optimization achieves 99% sensor coverage with faster convergence than Raindrop (82%), Grey Wolf (85%), Black Hole (89%), Dragonfly (89%), and standard Aquila Optimization (93%). Statistical analysis confirms the significant performance difference between the proposed and existing algorithms. Overall, the proposed approach provides a practical, scalable, and energy-efficient solution for sustainable wireless sensor network deployment.

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e18" altimg="si16.svg"> <mml:mi>t</mml:mi> </mml:math> -linear coloring in graphs with given small lenient-tree-width

Linear colorings of graphs

An incidence of a graph $G$ is a pair $(x,e)$, where $x$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $x$. Two incidences $(x,e)$ and $(y,f)$ are adjacent if any one of the following holds: (i) $x=y$, or (ii) $e=f$, or (iii) $xy=e$ or $f$. In [3], Brualdi and Massey introduced the concept of incidence coloring of a graph $G$ as a mapping from the set of incidences of $G$ to a finite set of colors such that adjacent incidences receive distinct colors. A signed graph $(G,\sigma)$ consists of a graph $G$ and the signature $\sigma : E(G)\rightarrow \{+1,-1\}$. In this paper, we define an incidence coloring of signed graphs as a natural generalization of the usual notion of incidence coloring of unsigned graphs. We prove that our definition is compatible with switching operation. We also prove that the incidence chromatic number (in signed sense) of a signed graph $(G,\sigma)$ coincide with the incidence chromatic number (in the usual unsigned sense) of its underlying graph $G$. The exact value or upper bounds which are known for the incidence chromatic numbers of some well-known families of unsigned graphs are also mentioned for their signed versions, namely, signed cycles, signed trees, signed complete graphs and signed toroidal grids.

Hardware implementations of the Ising model offer promising solutions to large-scale optimization tasks. In the literature, various nanodevices have been shown to emulate the spin dynamics for such Ising machines with remarkable effectiveness. Other nanodevices have been shown to implement spin-spin coupling with compact footprint and minimal energy dissipation. However, an ideal Ising machine would associate both types of nanodevices, and they must operate synergistically to support annealing: a progressive reduction of machine stochasticity that allows it to settle to an energy minimum. Here, we report an Ising machine that combines two nanotechnologies: memristor crossbar – storing multi-level couplings – and stochastic magnetic tunnel junction (SMTJ), acting as thermally driven spins. Because the same read voltage that interrogates the crossbar also biases the SMTJs, increasing this voltage automatically lowers the effective temperature of the machine, providing an intrinsic, analog-native annealing technique. Operating at zero magnetic field, our prototype consistently reaches the global optimum of a 24-vertex weighted MAX-CUT and a 10-vertex, three-color graph-coloring problem using an externally implemented feedback loop. Given that both nanotechnologies in our demonstrator are CMOS-integrated, this approach is compatible with advanced 3D integration, offering a scalable pathway toward compact, fast, and energy-efficient large-scale Ising solvers.

Graph Coloring is the process of assigning colors to vertices such that no adjacent vertices share the same color, using the minimal number of colors possible. This study aims to implement graph coloring to the map of North Gorontalo Regency, which consists of 11 sub-districts and 123 villages, utilizing the D’Satur and Backtracking algorithms. It also compares the algorithms in terms of the smallest chromatic number and identifies strategic points in North Gorontalo Regency, particularly in sub-districts, based on the number of adjacent vertices. The study employed a case study method to gather information, specifically the map of North Gorontalo Regency. The results demonstrate that graph coloring of the map utilizing the D’Satur algorithm produces a chromatic number of (χ = 3) for sub-districts and (χ = 5) for villages. Meanwhile, the Backtracking algorithm yields a chromatic number of (χ = 3) for districts and (χ = 4) for villages. Thus, for sub-district coloring, both algorithms yield the same chromatic number. However, the Backtracking algorithm performs better for village coloring, as it produces the smallest chromatic number. The identified strategic sub-district is Kwandang, which has the highest degree of 4.

Shameful Inequalities for List and DP Coloring of Graphs

Content-Based Image Retrieval (CBIR) systems critically depend on discriminative yet efficient feature representations to retrieve relevant images from large-scale databases. However, many existing handcrafted and graph-based methods face limitations in scalability and in jointly modeling multimodal information such as color, texture, and spatial relationships. To address these challenges, this paper proposes a novel feature extraction framework termed Multi-modal Graph Color SIFT (MGC-SIFT). In the proposed approach, color-augmented SIFT descriptors extracted in the YCbCr color space are organized as a graph of local keypoints, over which Graph Neural Networks (GNNs) are applied to model inter-keypoint spatial relationships. An attention mechanism is incorporated to emphasize discriminative keypoint regions, while proxy-based learning is employed to improve representation compactness and retrieval efficiency.The effectiveness of MGC-SIFT is evaluated on four benchmark datasets—Corel-1K, COIL-20, Oxford-102 Flowers, and UC-Merced Land Use—covering natural scenes, controlled object images, fine-grained categories, and aerial imagery. Experimental evaluation using standard CBIR metrics, including mean Average Precision (mAP), Precision@k, Recall@k, F1-score@k, and Accuracy@k, demonstrates that the proposed method achieves consistent and competitive retrieval performance across heterogeneous datasets, including robustness under image degradation conditions. Ablation studies further confirm the complementary contributions of color augmentation, graph-based modeling, attention mechanisms, and proxy-based learning. In addition, runtime and memory analysis indicate that proxy-based learning significantly reduces retrieval latency, supporting scalable image retrieval.Overall, the proposed MGC-SIFT framework provides a robust and interpretable multimodal representation for CBIR by explicitly modeling joint color–spatial dependencies at the local keypoint level, offering a practical solution for scalable image retrieval in real-world applications.

Approximate Quantum 3-Colorings of Graphs and the Quantum Max 3-Cut Problem

Polynomials counting group colorings in graphs

We consider the problem of feasible vertex coloring with the minimum number of colors for connected undirected graphs that contain no self-loops or multiple edges. For every given $k \geq 3$, the problem of checking the existence of a feasible vertex coloring of the graph with k colors is NP-complete. Therefore, studying graph-scaling processes while preserving or limiting their chromatic numbers is of interest. In this paper, we study the nature of changes in the chromatic number of graphs with an increase in the number of vertices and edges using gluing operations by identifying their isomorphic subgraphs. $G = (G_{1} \circ G_{2}) \tilde{G}$ — is the resulting graph of the gluing operation of graphs $G_1$ and $G_2$; $\tilde{G} \subseteq G$ is the subgraph obtained as a result of identifying isomorphic subgraphs $G_1' \subseteq G_1$ and $G_2' \subseteq G_2$; $|V(G)| = |V(G_1)| + |V(G_2)| - |V(\tilde{G})|, |E(G)| = |E(G_1)| + |E(G_2)| - |E(\tilde{G})|$. Gluing operations in which one of the graphs $G_1$ or $G_2$ is isomorphic to another graph or its subgraph and the identification of subgraphs $G_1^{'}\subset G_1$ and $G_2^{'} \subset G_2$ is carried out in accordance with the isomorphism $G_1' \cong G_2'$, are called cloning operations. A constructive description of a class of 2-chromatic graphs is obtained based on the gluing and cloning operations. Constraints on the gluing and cloning operations that ensure the preservation of the chromatic number of scalable graphs are formulated. It is established that when performing cloning operations, $\chi(G) =\max{\chi(G_1),\chi(G_2)}$. Examples of assembling 2-chromatic graphs using operations satisfying these constraints are given. For an arbitrary gluing operation $\chi(G) \leqslant \max {\chi(G_1),\chi(G_2)} + |V(\tilde{G})| - |V(\tilde{G'})|$, where $\tilde{G'}$ is the maximal complete subgraph of $\tilde{G}$. The possible growth of the chromatic number of graphs is estimated when scaling with various restrictions on the superposition of gluing operations.