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.
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-pol...
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, inclu...
Juan Gutiérrez, Grover Ugarte· Anais do XI Encontro de Teor...· 0 citations
A graph is
perfect
if, for every induced subgraph, the chromatic number equals the size of its largest clique. In 1972, Lovász established a fundamental characterization of perfect graphs, showing that a graph is perfect if and only if, for every induced subgraph, the product of the size of the largest independen...
The Petersen coloring conjecture of Jaeger asserts that every bridgeless cubic graph admits a Petersen coloring. Recently, Putman presented an explicit counterexample on $112$ vertices and verified its non-colorability by showing, using a SAT solver, that an instance with $3640$ variables and $68324$ clauses is unsatis...
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...
Ilkyoo Choi· Journal of Graph Theory· 1 citation· ⚡1
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
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