1. Paint cost spectrum of perfect k-ary treesSonwabile Mafunda, Jonathan L. Merzel, Katherine E. Perry, Anna Varvak, 2026, original scientific article 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: 566; Downloads: 8
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2. Brooks' type theorems for coloring parameters of locally finite graphs and Kőnig's LemmaAmitayu Banerjee, Zalán Molnár, Alexa Gopaulsingh, 2025, original scientific article 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: 874; Downloads: 6
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3. Adjacent vertex distinguishing total coloring of corona product of graphsHanna Furmańczyk, Rita Zuazua, 2025, original scientific article Abstract: An adjacent vertex distinguishing total k-coloring f of a graph G is a proper total k-coloring of G such that no pair of adjacent vertices has the same color sets, where the color set at a vertex v, C_f^G(v), is {f(v)} ∪ {f(vu)|u ∈ V(G), vu ∈ E(G)}. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph G has adjacent vertex distinguishing total (Δ(G) + 3)-coloring. In this paper we confirm the conjecture for many types of coronas, in particular for generalized, simple and l-coronas of graphs, not relating the results to particular graph classes of the factors.
Keywords: corona graph, l-corona, generalized corona graph, adjacent vertex distinguishing total coloring, AVDTC Conjecture
Published in RUP: 21.10.2025; Views: 785; Downloads: 5
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