Open Problems from the 32nd Workshop on Cycles and Colourings
This document presents a compilation of open problems in graph theory, specifically focusing on cycles and colorings, stemming from discussions at the 32nd Workshop on Cycles and Colourings held in Poprad, Slovakia.
Spectral Approaches for Relaxed Graph Colorings: Exploring d-Improper and t-Clustered Chromatic Numbers
This research paper explores the potential of using spectral graph theory to analyze and bound relaxed graph coloring problems, specifically focusing on d-improper and t-clustered colorings, and conjectures a close relationship between the chromatic number of a graph and the d-improper chromatic number of its strong product with a complete graph.
Palette Sparsification for Coloring Graphs with Sparse Neighborhoods
This paper presents a new palette sparsification theorem for efficiently coloring graphs with sparse neighborhoods, improving upon previous results and potentially leading to faster algorithms for this problem.
Edge Coloring of Split Graphs: Classifying a Subclass of (σ = 3)-Split Graphs
This research paper presents a polynomial-time algorithm to solve the edge coloring problem for a subclass of (σ = 3)-split graphs, contributing to the ongoing effort to fully classify split graphs based on their chromatic index.
On the Chromatic Number of Powers of Subdivisions of Graphs
This research paper presents an asymptotically tight upper bound for the chromatic number of the 3rd power of the 3rd subdivision of a graph G, denoted as G33, confirming a conjecture by Mozafari-Nia and Iradmusa in the case m=n=3.
Neighborhood Balanced 3-Coloring of Graphs
This research paper explores the concept of neighborhood balanced 3-coloring in graph theory, focusing on the conditions required for a graph to have this property and providing characterizations for specific graph families.
2-Distance 4-Coloring of Planar Subcubic Graphs: An Investigation of Girth Constraints
This research paper investigates the relationship between the girth (length of the shortest cycle) and the 2-distance chromatic number of planar subcubic graphs, aiming to determine the minimum girth required to guarantee a 2-distance 4-coloring.
Non-Conflicting Nowhere Zero Z2 × Z2 Flows in Cubic Graphs and Their Applications to Edge-Coloring
This paper introduces the concept of non-conflicting nowhere zero Z2 × Z2 flows in cubic graphs and demonstrates their application in proving the existence of normal 6-edge-colorings, particularly in claw-free bridgeless cubic graphs and those with 2-factors containing at most two cycles.
Strong Odd Colorings of Graphs: Exploring Bounds and Properties
This research paper investigates the properties and bounds of strong odd colorings, a new graph coloring concept, for various graph classes including trees, unicyclic graphs, planar graphs, outerplanar graphs, and graph products.
25-vertex triangle-free graph is 3-dicritical
The 25-vertex triangle-free graph is 3-dicritical, demonstrating the minimum size for a 3-dichromatic triangle-free graph.
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