A graph G is (d1,d2,d3)-colorable if the vertex set V(G) can be partitioned into three subsets V1,V2 and V3 such that for i∈{1,2,3} induced G[Vi] has maximum vertex-degree at most di. So, (0,0,0)-colorability exactly 3-colorability. The well-known Steinberg's conjecture states every planar without cycles of length 4 or 5 3-colorable. As this being disproved by Cohen-Addad etc. in 2017, a simila...

For 1≤s1≤s2≤…≤sk and a graph G, packing (s1,s2,…,sk)-coloring of G is partition V(G) into sets V1,V2,…,Vk such that, for each 1≤i≤k the distance between any two distinct x,y∈Vi at least si+1. The chromatic number, χp(G), smallest k that has (1,2,…,k)-coloring. It known there are trees maximum degree 4 subcubic graphs with arbitrarily large χp(G). Recently, was series papers on (s1,s2,…,sk)-colo...

A graph is (d1, . . . , dr )-colorable if its vertex set can be partitioned into r sets V1, . . . , Vr where themaximum degree of the graph induced by Vi is at most di for each i ∈ {1, . . . , r}. Let Gg denote the class of planar graphs with minimum cycle length at least g . We focus on graphs in G5 since for any d1 and d2, Montassier and Ochem constructed graphs in G4 that are not (d1, d2)-co...

Let d, k be any two positive integers with k > d > 0. We consider a k-coloring of a graph G such that the distance between each pair of vertices in the same color-class is at least d. Such graphs are said to be (k,d)-colorable. The object of this paper is to determine the maximum size of (k, 3)-colorable, (k, 4)-colorable, and (k, k 1 )-colorable graphs. Sharp results are obtained for both (k, ...

Using the most recent updated physics, calibrated solar models have been computed with the new thermonuclear reaction rates of NACRE, the recently available European compilation. Comparisons with models computed with the reaction rates of Caughlan & Fowler (1988) and of Adelberger et al. (1998) are made for global structure, expected neutrinos fluxes, chemical composition and sound speed profil...

Hugo Hadwiger proved that a graph that is not 3-colorable must have a K4minor and conjectured that a graph that is not k-colorable must have a Kk+1minor. By using the Hochstättler-Nešetřil definition for the chromatic number of an oriented matroid, we formulate a generalized version of Hadwiger’s conjecture that might hold for the class of oriented matroids. In particular, it is possible that e...

Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colorable. This a strengthening of the famous Hadwiger’s Conjecture. Geelen et al. proved $$O(t\sqrt {\log t} )$$ -colorable. Using methods present authors Postle recently developed for coloring graphs minor, we make first improvement on this bound by showing O(t(logt)β)-colorable β > 1/4.

We introduce the notion of complex chromatic number signed graphs as follows: given set $${\mathbb {C}}_{k,l}=\{\pm 1, \pm 2, \ldots , k\}\cup \{\pm 1i, 2i, li\}$$ where $$i=\sqrt{-1}$$ a graph $$(G,\sigma )$$ is said to be (k, l)-colorable if there exists mapping c vertices G {C}}_{k,l}$$ such that for every edge xy we have $$\begin{aligned} c(x)c(y)\ne \sigma (xy) |c(x)^2|. \end{aligned}$$ Th...