6‐Star‐Coloring of Subcubic Graphs

Abstract

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two adjacent vertices are assigned the same color) such that no path on four vertices is 2‐colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6‐star‐colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed (J Graph Theory 47(3) (2004), 140–153).

Star chromatic index of subcubic multigraphs

“…Then ( * ) ∈ ( 1 ) ∪ ( 2 ), otherwise we obtain a star 6-edge-coloring of by coloring the edge by color ( * ) and by a color in [6]∖( ( * ) ∪ ( )), a contradiction. We next show that ( 1 ) ∪ ( 2 ) = [6]. Suppose that ( 1 ) ∪ ( 2 ) ≠ [6].…”

“…We next show that ( 1 ) ∪ ( 2 ) = [6]. Suppose that ( 1 ) ∪ ( 2 ) ≠ [6]. Now coloring the edge by a color, say , in [6]∖( ( 1 ) ∪ ( 2 )), and then coloring by color ( 1 ) if ( 1 ) ∉ ( * ) or a color in…”

“…, we obtain a star 6-edge-coloring of , a contradiction. Thus ( 1 ) ∪ ( 2 ) = [6] and so ( 1 ) = ( 2 ) = 3. By symmetry, ( * 1 ) = ( * 2 ) = 3.…”

“…Let ∶ ( ∖ ) → [6] be a star edge-coloring of ∖ . We obtain a star 6-edgecoloring of by coloring the edges 1 by two distinct colors, say , , in [6]∖( ( * 1 ) ∪ { ( 3 * 3 )}) and 3 by a color in [6]∖( ( * 3 ) ∪ { , }), a contradiction. This proves that 1 , 2 , 3 are pairwise distinct.…”

Star 5-edge-colorings of subcubic multigraphs

“…As pointed out in [6], the definition of star edge-coloring of a graph G is equivalent to the star vertexcoloring of its line graph L(G). Star edge-coloring of a graph was initiated by Liu and Deng [10], motivated by the vertex version (see [1,4,5,8,11]). Given a multigraph G, we use |G| to denote the number of vertices, e(G) the number of edges, δ(G) the minimum degree, and ∆(G) the maximum degree of G, respectively.…”