A proper conflict-free coloring is a proper vertex coloring in which every nonisolated vertex has a color occurring uniquely in its open neighborhood. We prove that every graph with neither a $K_5$-minor nor a $Q_6$-minor admits such a coloring with at most seven colors, where $Q_6=K_3\vee\overline{K_3}$. In particular, this improves the previous general upper bound of eight for planar graphs. The proof combines a previously developed iterated distance-three selector construction with a general anchor-contraction lifting principle. The first supplies independently colored witnesses in closed neighborhoods, while the second combines those witnesses with a proper coloring of a suitable minor. We also develop the parity analogue of the first mechanism and show that, whenever the $K_{k+1}$ case of Hadwiger's conjecture holds, every $K_{k+1}$-minor-free graph can be proper vertex colored with $2k-1$ colors such that every nonisolated vertex has a color occurring an odd number of times in its open neighborhood.
A proper conflict-free coloring is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its open neighborhood. We prove that every finite simple graph with girth at least 7 and maximum average degree less than 8/3 admits a proper conflict-free coloring from arbitrary vertex lists of size at least the vertex degree plus 2. Consequently, every planar graph of girth at least 8 is proper conflict-free (degree+2)-choosable, improving the sufficient girth bound of 9 obtained from the earlier 18/7 maximum-average-degree theorem. The proof uses local extension lemmas for short threads, including threads with a common boundary endpoint. Two-element control sets and an incidence count yield a weighted thread inequality, which supplies the required bound on the charge sent by each vertex in a discharging argument.
A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furma\'nczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balab\'an [MFCS 2026].
It is proved that PCF-COLORABILITY is NP-complete for bipartite graphs, and linear-time algorithms for PCF-COLORABILITY are provided in block graphs, proper interval graphs, chain graphs, and pseudo-split graphs.
A strong edge coloring is a proper edge coloring in which every color class is an induced matching; the least number of colors is the strong chromatic index $\chi'_s(G)$. Lin and Lin proved that every claw-free subcubic graph other than the triangular prism satisfies $\chi'_s(G) \le 7$, with all their tight examples containing diamonds. Kardos (Problem 4.1 of the open-problem collection of the 33rd Workshop on Cycles and Colourings) asked whether every diamond-free claw-free cubic graph is strongly 6-edge-colorable, equivalently whether $\chi'_s(T(G))=6$ for every cubic graph $G$, where $T(G)$ is the truncation of $G$. We exhibit an explicit connected, simple, diamond-free, claw-free cubic graph $H$ on 18 vertices with $\chi'_s(H)=7$, and show that it has the fewest vertices possible for such a non-prism example. Hence the first formulation, over simple cubic graphs, is false even after excluding the prism; and since $H$ is the truncation of a cubic multigraph with parallel edges, the two formulations are not equivalent unless"cubic graph"is allowed to mean loopless multigraph, under which reading the problem is answered negatively. The narrower version restricted to truncations of simple base graphs remains open.
Given a graph $G=(V,E)$, a (proper) $k$-coloring for $G$ is a vertex coloring with $k$ colors such that every two adjacent vertices receive different colors. Suppose that the vertex set $V$ is partitioned into some groups, a proper coloring is called fair if for every color class, the difference between the number of vertices in any two groups does not exceed a given threshold. In this paper, we investigate the parameterized complexity of the fair coloring problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the number of groups for forests and also graphs of modular-width two, even when the number of colors is equal to two. On the positive side, we prove that when the number of colors is equal to two, then the problem is FPT with respect to neighborhood diversity of the input graph. Moreover, in general, the problem is FPT with respect to neighborhood diversity and the number of groups. As a by-product, we prove that unary vector bin packing problem is W[1]-hard with respect to the dimension.
R. Javadi, Hossein Shokouhi· arXiv.org· 0 citations
A graph $G$ is \emph{apex} if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.
Yuta Inoue, K. Kawarabayashi, Rintaro Matsuo et al.· 0 citations
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