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 ().
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...
This paper investigates the parameterized complexity of the fair coloring problem with respect to the structural parameters of the input graph and proves 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.
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...
A. Jiménez, C. Lintzmayer, M. Sambinelli· 1 citation
We introduce and begin the study of sequence b-colorings, a natural generalization of the classical notion of b-colorings introduced by Irving and Manlove in 1999. In a sequence b-coloring, each color class is required to contain a prescribed minimum number of color-dominating vertices (CDVs). We establish several fund...
Some equity constraints on the coloring classes of a classical coloring of the vertices of a graph give rise to the equitable coloring: the number of vertices colored with each color differs by at most one. The least number of colors for which a graph has such an equitable coloring is called the equitable chromatic num...
E. F. Olariu, C. Frăsinaru· INFORMS journal on computing· 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*-...
Jakub Balabán, Oliver Bukor· 0 citations
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