Jianliang Wu

On edge colorings of 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that every 1-planar graph with maximum degree Δ?10 can be edge-colored with Δ colors. Research highlights? In the study we investigate the edge coloring of 1-planar graphs. ? We describe an associated plane graph of 1-planar and use the discharging method in the detail proofs. ? It is proved that every 1-planar graphs with maximum degree at least 10 is of class 1 in terms of the edge coloring.

On the linear arboricity of planar graphs

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. Akiyama, Exoo, and Harary conjectured that [Δ/2] ≤ la(G) ≤ [Δ+1/2] for any simple graph G with maximum degree Δ. The conjecture has been proved to be true for graphs having Δ = 1, 2, 3, 4, 5, 6, 8, 10. Combining these results, we prove in the article that the conjecture is true for planar graphs having Δ(G) ≠ 7. Several related results assuming some conditions on the girth are obtained as well.

Neighbor sum distinguishing total colorings of planar graphs with maximum degree Δ

A (proper) total k]-coloring of a graph G is a mapping ? : V ( G ) ? E ( G ) ? k ] = { 1 , 2 , ? , k } such that any two adjacent elements in V ( G ) ? E ( G ) receive different colors. Let f ( v ) denote the sum of the color of a vertex v and the colors of all incident edges of v . A total k ] -neighbor sum distinguishing-coloring of G is a total k ] -coloring of G such that for each edge u v ? E ( G ) , f ( u ) ? f ( v ) . By ? n s d ? ( G ) , we denote the smallest value k in such a coloring of G . In this paper, we show that if G is a planar graph with Δ ( G ) ? 14 , then ? n s d ? ( G ) ? Δ ( G ) + 2 .

A Planar linear arboricity conjecture

The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [Math Slovaca 30 (1980), 405–417] stated the Linear Arboricity Conjecture (LAC) that the linear arboricity of any simple graph of maximum degree Δ is either ⌈Δ/2⌉ or ⌈(Δ + 1)/2⌉. In [J. L. Wu, J Graph Theory 31 (1999), 129–134; J. L. Wu and Y. W. Wu, J Graph Theory 58(3) (2008), 210–220], it was proven that LAC holds for all planar graphs. LAC implies that for Δ odd, la(G) = ⌈Δ/2⌉. We conjecture that for planar graphs, this equality is true also for any even Δ⩾6. In this article we show that it is true for any even Δ⩾10, leaving open only the cases Δ = 6, 8. We present also an O(n logn) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when Δ⩾9.

The Linear Arboricity of Series-Parallel Graphs

Abstract. The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. A graph is called series-parallel if it contains no subgraphs homeomorphic to K4. In this paper, we prove that for any series-parallel graph G having Δ (G)≥3. Since an outerplanar graph is a series-parallel graph, this is also true for any outerplanar graph.

On the neighbor sum distinguishing total coloring of planar graphs

Let c be a proper total coloring of a graph G = ( V , E ) with integers 1 , 2 , ? , k . For any vertex v ? V ( G ) , let ? c ( v ) denote the sum of colors of the edges incident with v and the color of v. If for each edge u v ? E ( G ) , ? c ( u ) ? ? c ( v ) , then such a total coloring is said to be neighbor sum distinguishing. The least k for which such a coloring of G exists is called the neighbor sum distinguishing total chromatic number and denoted by ? Σ ? ( G ) . Pil?niak and Wo?niak conjectured ? Σ ? ( G ) ? Δ ( G ) + 3 for any simple graph with maximum degree Δ ( G ) . It is known that this conjecture holds for any planar graph with Δ ( G ) ? 13 . In this paper, we prove that for any planar graph, ? Σ ? ( G ) ? max ? { Δ ( G ) + 3 , 14 } .

Some Results on List Total Colorings of Planar Graphs

Let G be a planar graph with maximum degree Δ. In this paper, it is proved that if Δ ? 9, then G is total-(Δ + 2)-choosable. Some results on list total coloring of G without cycles of specific lengths are given.

Minimum total coloring of planar graph

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of “total coloring”. Total coloring is a coloring of

Acyclic chromatic index of planar graphs with triangles

V\cup {E}