Yiqiao Wang

Acyclic Chromatic Indices of Planar Graphs with Girth At Least 4

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamik (Math. Slovaca 28 (1978), 139–145) and later Alon et al. (J Graph Theory 37 (2001), 157–167) conjectured that for any simple graph G with maximum degree Δ. In this article, we confirm this conjecture for planar graphs of girth at least 4.

Adjacent vertex distinguishing colorings by sum of sparse graphs

A neighbor sum distinguishing edge- k -coloring, or nsd- k -coloring for short, of a graph G is a proper edge coloring of G with elements from { 1 , 2 , ? , k } such that no pair of adjacent vertices meets the same sum of colors of G . The definition of this coloring makes sense for graphs containing no isolated edges (we call such graphs normal). Let mad ( G ) and Δ ( G ) be the maximum average degree and the maximum degree of a graph G , respectively. In this paper, we prove that every normal graph with Δ ( G ) ? 5 and mad ( G ) < 3 admits an nsd- ( Δ ( G ) + 2 ) -coloring. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

An improved upper bound on the linear 2-arboricity of planar graphs

The linear 2-arboricity la 2 ( G ) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. In this paper, we prove that if G is a planar graph, then la 2 ( G ) ? ? ( Δ ( G ) + 1 ) / 2 ? + 6 . This improves a result in Lih et?al. (2003), which says that every planar graph G satisfies la 2 ( G ) ? ? ( Δ ( G ) + 1 ) / 2 ? + 12 .

Planar Graphs with

A plane graph

\Delta\ge 9

G

are Entirely

is entirely

(\Delta+2)

k

-Colorable

-colorable if

The entire chromatic number of graphs embedded on the torus with large maximum degree

V(G)\cup E(G) \cup F(G)

Neighbor-sum-distinguishing edge choosability of subcubic graphs

can be colored with