104 research outputs found
Strong edge-colorings for k-degenerate graphs
We prove that the strong chromatic index for each -degenerate graph with
maximum degree is at most
Linear Choosability of Sparse Graphs
We study the linear list chromatic number, denoted \lcl(G), of sparse
graphs. The maximum average degree of a graph , denoted \mad(G), is the
maximum of the average degrees of all subgraphs of . It is clear that any
graph with maximum degree satisfies \lcl(G)\ge
\ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if
\mad(G)<12/5 and , then \lcl(G)=\ceil{\Delta(G)/2}+1, and
we give an infinite family of examples to show that this result is best
possible; (2) if \mad(G)<3 and , then
\lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to
show that the bound on \mad(G) cannot be increased in general; (3) if is
planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure
Linear colorings of subcubic graphs
A linear coloring of a graph is a proper coloring of the vertices of the
graph so that each pair of color classes induce a union of disjoint paths. In
this paper, we prove that for every connected graph with maximum degree at most
three and every assignment of lists of size four to the vertices of the graph,
there exists a linear coloring such that the color of each vertex belongs to
the list assigned to that vertex and the neighbors of every degree-two vertex
receive different colors, unless the graph is or . This confirms
a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is
constructive and yields a linear-time algorithm to find such a coloring
- β¦