List-coloring, introduced independently by Vizing and by Erd\H{o}s, Rubin, and Taylor in the 1970s, generalizes ordinary vertex coloring by assigning to each vertex its own set of admissible colors. A graph is chromatic-choosable if its list chromatic number equals its chromatic number. The previous survey on list-coloring by D R Woodall (2001), emphasized defective choosability, the list-coloring conjectures, and different methods used for list-coloring. This survey reviews major developments on list-coloring and chromatic-choosability, with emphasis on graph classes for which equality is known, graph classes exhibiting a nontrivial gap, and the principal methods used to prove such results. The survey covers embedded graphs, perfect graphs, complete bipartite and multipartite graphs, claw-free graphs, line graphs, powers of graphs, graph products, and selected variants of list-coloring.
The locating-chromatic number of a graph combines proper vertex coloring with vertex identification through distances to color classes. Although this parameter has been studied for many graph families, general results for bipartite graphs remain limited. Bipartite graphs contain structural symmetries, especially within each partite set, making it difficult to obtain distinct color codes. This paper establishes lower and upper bounds for the locating-chromatic number of bipartite graphs using neighborhood equivalence classes in the two partite sets. The bounds describe the effect of identical neighborhoods on the number of distinguishable color codes. They are shown to be tight, and regular complete bipartite graphs are identified as extremal examples. The paper also considers corona products of regular complete bipartite graphs and complements of complete graphs, for which bipartiteness is preserved. Exact values of the locating-chromatic number are obtained for all relevant numbers of attached vertices, indicating how the number and arrangement of pendant vertices affect the coloring process. These results provide a basis for studying locating colorings in bipartite and corona graphs and complement existing results in the literature.
Dian Kastika Syofyan, E. Baskoro, H. Assiyatun et al.· Baghdad Science Journal· 0 citations
An r-dynamic coloring of a graph G is a proper vertex coloring in which each vertex sees at least min{r, d(v)} distinct colors in its neighborhood. The minimum number of colors in such a coloring is the r-dynamic chromatic number χdr(G). We determine exact values and upper bounds of χdr for several graph classes, including triangular grids, planar 3-trees for r ≤ 4, and planar Eulerian triangulations for r ≤ 3 (with a partial result for r = 4), confirming the conjecture of [Song et al. 2014] for these subclasses. We also establish exact values for the 2-dynamic chromatic number of a subclass of circulant graphs, confirming a conjecture of [Montgomery 2001] for this regular family.
Juan Gutiérrez, Grover Ugarte· Anais do XI Encontro de Teor...· 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].
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.
Richard Bi, Peter Bradshaw· Journal of Graph Theory· 0 citations
We give an explicit simple bridgeless cubic graph on 112 vertices with no Petersen coloring, and hence no normal 5-edge-coloring. The graph is identified by the SHA-256 digest in Theorem 1.1. It is assembled from three copies of a four-pole L and a claw six-pole C; in turn, L is assembled from four copies of a four-pole F and one copy of C, where F is obtained from the Petersen graph by deleting the endpoints of one edge. We give direct SAT formulations for Petersen colorings and normal 5-edge-colorings. CaDiCaL 3.0.1 returned UNSAT for both formulas, and drat-trim verified the resulting DRAT proofs. The ancillary archive contains the construction, an explicit relabeling, the encoders, certificates, hashes, and verification programs. Combined with a theorem of Ma, Mattiolo, Steffen, and Wolf, the counterexample also implies that infinitely many connected simple bridgeless cubic graphs have no Petersen coloring. We also give a separately verified, nonisomorphic $D_3$-symmetric 112-vertex counterexample. We do not address whether 112 is minimum.
Recently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 186 by refining their proof. The original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction of t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.