Wensong Lin

Several parameters of generalized Mycielskians

The generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m ≥ 0, one can transform G into a new graph µm (G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of µm (G).

On 2-distance coloring of plane graphs with girth 5

A vertex coloring is called 2 -distance if any two vertices at distance at most 2 from each other get different colors. Let ź 2 ( G ) be the 2-distance chromatic number of a graph G . Suppose G is a plane graph with girth 5 and maximum degree Δ . In this paper, we prove that if Δ ź { 7 , 8 } , then ź 2 ( G ) ź Δ + 7 . Furthermore, we show that ź 2 ( G ) ź Δ + 4 if Δ is sufficiently large.

Entire coloring of plane graph with maximum degree eleven

A plane graph is called entirely k -colorable if for each x ? V ( G ) ? E ( G ) ? F ( G ) , we can use k colors to assign each element x a color such that any two elements that are adjacent or incident receive distinct colors. In this paper, we prove that if G is a plane graph with Δ = 11 , then G is entirely ( Δ + 2 ) -colorable, which provides a positive answer to a problem posed by Borodin (Problem 5.2 in Borodin (2013)).

On linear coloring of planar graphs with small girth

Abstract A vertex coloring of a graph G is linear if the subgraph induced by the vertices of any two color classes is the union of vertex-disjoint paths. In this paper, we study the linear coloring of graphs with small girth and prove that: (1) Every planar graph with maximum degree Δ ≥ 39 and girth g ≥ 6 is linearly ( ⌈ Δ 2 ⌉ + 1 ) -colorable. (2) There exists an integer Δ 0 such that every planar graph with maximum degree Δ ≥ Δ 0 and girth g ≥ 5 is linearly ( ⌈ Δ 2 ⌉ + 1 ) -colorable. The latter result is best possible in some sense.