Feb 6, 2014 � For a fixed integer r > 0 , the r -hued chromatic number of G , denoted by χ r ( G ) , is the smallest k such that G has a ( k , r ) -coloring.

A list assignment L of G is a mapping that assigns every vertex v ∈ V(G) a set L(v) of pos- itive integers. For a given list assignment L of G, an (L,�...

Mar 15, 2022 � For a given list assignment L of a graph G , an ( L , r ) -coloring of G is a proper coloring c such that for any vertex v with degree d ( v )�...

Let K(r)=r+3 if 2≤r≤3, and K(r)=�ŒŠ3r/2�Œ+1 if r≥4. We proved that if G is a K4-minor free graph, thenχr(G)≤K(r), and the bound can be attained;χL,r(G)≤K(r)+1.

The rr-hued chromatic number of GG, χr(G)χr(G), is the least integer kk such that for any v∈V(G)v∈V(G) with L(v)={1,2,…,k}L(v)={1,2,…,k}, GG has an (L,r)(L,r)-�...

The decomposi- tions are applied to show that if G is a K4(7)-minor free graph, then χr(G) ≤ f(r) if and only if G is not isomorphic to K6. � 2020 Elsevier Inc.

Mar 15, 2022 � In Song et al. (2014), it is proved that if G is a K 4-minor-free graph, then χ L, r ( G ) ≤ K ( r ) + 1. Let K 4 ( n ) be the set of all�...

The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k, r)-coloring. Let f(r)=r+3 if 1 ≤ r ≤ 2, f(r)=r+5 if 3 ≤ r�...

A strong edge coloring of a graph G is a proper coloring of edges in G such that any two edges of distance at most 2 are colored with distinct colors.

TL;DR: In this paper , the upper bound of the list r-hued chromatic number for all outer-1-planar graphs was shown for any positive integer r, where r is the�...