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Povzetek: A ▫${\ strong edge-coloring}$▫ of a graph is a proper edge-coloring, in which the edges of every path of length ▫$3$▫ receive distinct colors; in other words, every pair of edges at distance at most ▫$2$▫ must be colored differently. The least number of colors needed for a strong edge-coloring of a graph is the ▫${\ strong chromatic index}$▫. We consider the list version of the coloring and prove that the list strong chromatic index of graphs with maximum degree ▫$3$▫ is at most ▫$10$▫. This bound is tight and improves the previous bound of ▫$11$▫ colors. We also consider the question whether the strong chromatic index and the list strong chromatic index always coincide. We answer it in negative by presenting an infinite family of graphs for which the two invariants differ. For the special case of the Petersen graph, we show that its list strong chromatic index equals ▫$7$▫, while its strong chromatic index is ▫$5$▫. Up to our best knowledge, this is the first known edge-coloring for which there are graphs with distinct values of the chromatic index and its list version. In relation to the above, we also initiate the study of the list version of the normal edge-coloring. A ▫${\ normal edge-coloring}$▫ of a cubic graph is a proper edge-coloring, in which every edge is adjacent to edges colored with ▫$4$▫ distinct colors or to edges colored with ▫$2$▫ distinct colors. It is conjectured that ▫$5$▫ colors suffice for a normal edge-coloring of any bridgeless cubic graph and this statement is equivalent to the Petersen Coloring Conjecture. It turns out that similarly to strong edge-coloring, list normal edge-coloring is much more restrictive and consequently for many graphs the list normal chromatic index is greater than the normal chromatic index. In particular, we show that there are cubic graphs with list normal chromatic index at least ▫$9$▫, there are bridgeless cubic graphs with its value at least ▫$8$▫, and there are cyclically ▫$4$▫-edge-connected cubic graphs with value at least ▫$7$▫.
Ključne besede: strong edge-coloring, list strong edge-coloring, normal edge-coloring, list normal edge-coloring, Petersen coloring, Petersen coloring conjecture
Objavljeno v DiRROS: 05.02.2026; Ogledov: 177; Prenosov: 51
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Povzetek: A proper coloring ▫$\phi$▫ 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 ▫$|\phi^{-1}(c) \cap N_G(v)| = 1$▫. 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 ▫$|L(v)| \ge d_G(v) + k$▫ for every vertex ▫$v \in V(G)$▫, ▫$G$▫ admits a proper conflict-free coloring ▫$\phi$▫ such that ▫$\phi(v) \in L(v)$▫ for every vertex ▫$v \in V(G)$▫. 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.
Ključne besede: proper conflict-free coloring, list coloring, degree-choosability, degeneracy
Objavljeno v DiRROS: 05.02.2026; Ogledov: 120; Prenosov: 88
Celotno besedilo (495,29 KB)
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Povzetek: The subpath number of a graph ▫$G$▫ is defined as the total number of subpaths in ▫$G$▫, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper Knor et al. (Knor M, Sedlar J, Škrekovski R, et al (2026) Invitation to the subpath number[J]. Appl Math Comput 509:129646), where we investigated the subpath number and identified extremal graphs within the classes of trees, unicyclic graphs, bipartite graphs, and cycle chains. Here, we focus on the subpath number of cactus graphs and characterize all maximal and minimal cacti with ▫$n$▫ vertices and ▫$k$▫ cycles. We prove that maximal cacti are cycle chains in which all interior cycles are triangles, while the two end-cycles differ in length by at most one. In contrast, the minimal cacti consist of ▫$k$▫ cycles, all of which are end-triangles, with the subgraph induced by the remaining vertices forming a forest. By comparing extremal cacti with respect to the subpath number to those that are extremal for the subtree number and the Wiener index, we demonstrate that the subpath number does not correlate with either of these quantities, as their corresponding extremal graphs differ.
Ključne besede: subpath number, cactus graphs, extremal graphs, cycle chains, Wiener index, subtree number
Objavljeno v DiRROS: 05.02.2026; Ogledov: 124; Prenosov: 80
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Povzetek: Measuring sustainability as an efficient tool to achieve sustainable development and improve economic, social, and environmental aspects is always fraught with complications. In this sense, developing a suitable approach for evaluating and recognizing the strengths and weaknesses across these dimensions is paramount. Given the inherent uncertainty in data for many real-world applications, the primary aim of this paper is to present a data envelopment analysis (DEA) model for evaluating sustainability within a stochastic environment. The proposed model is non-radial and incorporates undesirable outputs, enabling the assessment of overall sustainability as well as each of the economic, social, and environmental dimensions simultaneously. This multi-dimensional evaluation capability is a key advantage of the proposed model. Additionally, the proposed model is based on input excesses and output shortfalls. Another notable advantage is the incorporation of the assumption of managerial disposability when dealing with undesirable outputs. To demonstrate the applicability of the proposed model, data from 59 diverse countries across Africa, Europe, North America, and Asia were analyzed over a 12-year period (2010–2022). The country selection was designed to capture global heterogeneity in development levels, policies, and environmental conditions, allowing for robust cross-continental comparisons. Key findings reveal that: (1) Europe achieves the highest stochastic sustainability scores, while North America performs poorest; (2) Environmental sustainability shows the most success cases globally, whereas social sustainability lags; (3) Significant trade-offs exist between economic growth and environmental protection.
Ključne besede: data envelopment analysis, performance analytics, Stochastic modeling, sustainability efficiency, environmental benchmarking, managerial disposability
Objavljeno v DiRROS: 04.02.2026; Ogledov: 136; Prenosov: 50
Celotno besedilo (1,45 MB)

Povzetek: The role of data centers has intensified with the expansion of the digital economy and the advancement of information and communication technologies. Their environmental footprint is determined by the electricity mix, whose temporal and spatial variability is insufficiently addressed in the conventional life cycle assessment (LCA). In this study, a time-resolved environmental impact assessment was applied to electricity generation in Slovenia and Serbia in 2023. The focus was on three categories: climate change, resource use (minerals and metals), and water use. Hourly generation data from the ENTSO-E Transparency platform were linked with the Ecoinvent 3.11 datasets to generate hourly impact profiles and representative daily profiles for summer and winter. The study's results reveal clear differences primarily due to the distinct electricity mix structures of the two countries. Slovenia relies on nuclear, hydro, and photovoltaic power, while Serbia is predominantly coal-based. Photovoltaic generation in Slovenia reduces greenhouse gas emissions during daylight but increases the impacts related to the use of minerals and metals. Serbia exhibits higher climate change burdens yet lower variability in other categories. Seasonal and diurnal fluctuations influence emission intensities, underscoring the limits of static, annualized assessments. The findings provide input for policy and smart infrastructure planning. Strategies for electric vehicle charging, data centers, and demand-side measures should integrate temporal profiles of environmental impacts. Tools such as environmentally differentiated tariffs or time-varying carbon pricing can help align energy use with periods of lower impact. More broadly, the results highlight trade-offs between greenhouse gas mitigation and other pressures, underscoring the need for holistic energy transition pathways.
Ključne besede: data centers, life cycle assessment, electricity mix, climate change, temporal variability
Objavljeno v DiRROS: 03.02.2026; Ogledov: 146; Prenosov: 67
Celotno besedilo (12,76 MB)
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