Jul 2026· Journal of Graph Theory· 0 citations· 14 references
Abstract
In the
flexible list coloring
problem, we consider a graph and a color list assignment on , as well as a subset for which each has a preferred color . Our goal is to find a proper ‐coloring of such that for at least vertices . We say that is ‐flexibly ‐choosable if for every ‐size list assignment on and every subset of vertices with coloring preferences, has a proper ‐coloring that satisfies an proportion of these coloring preferences. Dvořák, Norin, and Postle [Journal of Graph Theory, 2019] asked whether every ‐degenerate graph is ‐flexibly ‐choosable for some constant . In this paper, we prove that there exists a constant such that every graph with maximum average degree less than 3 is ‐flexibly 3‐choosable, which gives a large class of 2‐degenerate graphs which are ‐flexibly ‐choosable. In particular, our results imply a theorem of Dvořák, Masařík, Musílek, and Pangrác [Journal of Graph Theory, 2020] stating that every planar graph of girth 6 is ‐flexibly 3‐choosable for some constant . To prove our result, we generalize the existing reducible subgraph framework traditionally used for flexible list coloring to allow reducible subgraphs of arbitrarily large order.
A graph has an
‐coloring
if there exists an assignment from the vertices to subsets of with size such that adjacent vertices are assigned disjoint subsets. Odd girth at least is a necessary condition for a graph to have a ‐coloring. Chen and Raspaud conjectured a tight upper bound on the maximum average degree of a graph with odd girth at least that guarantees a ‐coloring. Namely, they conjectured that every graph with odd girth at least and maximum average degree less than has a ‐coloring. This conjecture is true for ; when , computers were used to perform case analysis. The main result of this paper confirms the conjecture for the next open case () without the use of computers. Moreover, our approach yields simpler, computer‐free proofs for previously known cases ().
Ilkyoo Choi· Journal of Graph Theory· 1 citation· ⚡1
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 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. Jiménez, C. Lintzmayer, M. Sambinelli· 1 citation
We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $\chi_\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $\chi_\mathrm{stable}(G)$. This shows that $\chi_\mathrm{stable}(G)$ is well-defined. We also show that $O(t \log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.
Tomohiro Koana, Y. Oh, Hirotaka Yoneda· 0 citations
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].
A total coloring of a graph is an assignment of colors to its vertices and edges so that adjacent or incident elements receive distinct colors, and it is equitable when the cardinalities of any two color classes differ by at most one. Stemock conjectured that every $4$-total coloring of a cubic graph of order less than $20$ is equitable. In this paper, we disprove this conjecture: the circular ladder $L_{12}$ admits a non-equitable $4$-total coloring and, moreover, no smaller counterexample exists: order $4$ is vacuous, and every $4$-total coloring of a cubic graph of order $6$, $8$, or $10$ is equitable. We also prove that the same property holds at order $14$. Our proofs rely on a decomposition lemma, which states that, in any $4$-total coloring of a cubic graph $G$, each color class consists of an independent set $S$ together with a perfect matching of $G-S$. We use the lemma to determine all possible color class configurations for orders $12$, $16$, and $18$, and we show that every listed configuration is attained. Finally, we provide a splicing construction showing that, for every even $n\geq16$, some connected cubic graph of order $n$ admits a non-equitable $4$-total coloring. We may conclude that $14$ is the largest order for which every $4$-total coloring of every cubic graph is equitable.
Matheus Adauto, C. D. de Figueiredo, Diana Sasaki et al.· 0 citations
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