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Abstract: We study reflection-symmetric realisations of symmetric graphs in the plane that allow a continuous symmetry and edge-length preserving deformation. To do so, we identify a necessary combinatorial condition on graphs with reflection-symmetric flexible realisations. This condition is based on a specific type of edge colouring, where edges are assigned one of three colours in a symmetric way. From some of these colourings we also construct concrete reflection-symmetric realisations with their corresponding symmetry preserving motion. We study also a specific class of reflection-symmetric realisations consisting of triangles and parallelograms.
Keywords: Flexible framework, rigidity, reflection symmetry, edge coloring
Published in RUP: 18.08.2026; Views: 13; Downloads: 1
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Abstract: We determine the paint cost spectrum for perfect k-ary trees. A coloring of the vertices of a graph G with d colors is said to be d-distinguishing if only the trivial automorphism preserves the color classes. The smallest such d is the distinguishing number of G and is denoted Dist(G). The paint cost of d-distinguishing G, denoted ρd(G), is the minimum size of the complement of a color class over all d-distinguishing colorings. A subset S of the vertices of G is said to be a fixing set for G if the only automorphsim that fixes the vertices in S pointwise is the trivial automorphism. The cardinality of a smallest fixing set is denoted Fix(G). In this paper, we explore the breaking of symmetry in perfect k-ary trees by investigating what we define as the paint cost spectrum of a graph G: (Dist(G); ρDist(G)(G), ρDist(G)+1(G), . . . , ρFix(G)+1(G)) and the paint cost ratio of G, which is defined to be the fraction of paint costs in the paint cost spectrum equal to Fix(G). We determine both the paint cost spectrum and the paint cost ratio completely for perfect k-ary trees. We also prove a lemma that is of interest in its own right: given an n-tuple, n ≥ 2 of distinct elements of an ordered abelian group and 1 ≤ k ≤ n! − 1, there exists a k × n row permuted matrix with distinct column sums.
Keywords: distinguishing coloring, fixing set, symmetry, cost of distinguishing
Published in RUP: 23.03.2026; Views: 583; Downloads: 8
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Abstract: In the past, analogues to Brooks’ theorem have been found for various parameters of graph coloring for infinite locally finite connected graphs in ZFC. We prove that there is a model of ZF (i.e., the Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) where these theorems fail. Moreover, such theorems follow from Kőnig’s Lemma (every infinite locally finite connected graph has a ray–a weak form of AC) in ZF. In ZF, inspired by a combinatorial argument of Herrlich and Tachtsis from 2006, we formulate new conditions for the existence of the distinguishing chromatic number, the distinguishing chromatic index, the total chromatic number, the total distinguishing chromatic number, the odd chromatic number, and the neighbor-distinguishing index in infinite locally finite connected graphs, which are equivalent to Kőnig’s Lemma. In this direction, we strengthen a recent result of Stawiski from 2023. We also generalize an algorithm of Imrich, Kalinowski, Pilśniak, and Shekarriz to show that the statement “If G is a connected infinite graph where the maximum degree Δ(G) ≥ 3 is finite, then the list-distinguishing chromatic number is at most 2Δ(G) − 1” holds under Kőnig’s Lemma in ZF. However, we prove that there is a model of ZF where the above statement fails.
Keywords: Axiom of Choice, Kőnig's Lemma, Brooks’ theorem, distinguishing proper coloring, total coloring, list-distinguishing proper coloring
Published in RUP: 22.10.2025; Views: 884; Downloads: 6
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Keywords: reconfiguration, dynamic programming, graph coloring
Published in RUP: 26.10.2021; Views: 4593; Downloads: 13
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Keywords: graph coloring, Latin square graph, pandiagonal Latin square graph, Cayley graph, hypergraph
Published in RUP: 17.09.2018; Views: 3978; Downloads: 47
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