Zhengke Miao

Optimal Channel Assignment for Wireless Networks Modelled as Hexagonal and Square Grids

Wireless networks are often modelled as different grids and the channel assignment problem for interference avoidance is formulated as a coloring problem of the grid graph, where channels (i.e., colors) assigned to interfering stations (i.e., vertices) at distance i must be at least δi apart, while the same channel can be reused in vertices whose distance is at least σ. In this paper, we consider the channel assignment problem for wireless networks modelled as hexagonal and square grids. We present optimal channel assignment algorithms for the case where the co-channel reuse distance σ is 4 and the minimum channel separation δi is σ-i

Adjacent vertex distinguishing total coloring of graphs with maximum degree 4

Abstract Let k be a positive integer. An adjacent vertex distinguishing (for short, AVD) total k -coloring ϕ of a graph G is a proper total k -coloring of G such that no pair of adjacent vertices have the same set of colors, where the set of colors at a vertex v is { ϕ ( v ) } ∪ { ϕ ( e ) : e xa0isxa0incidentxa0toxa0 v } . Zhang et al. conjectured in 2005 that every graph with maximum degree Δ has an AVD total ( Δ + 3 ) -coloring. Recently, Papaioannou and Raftopoulou confirmed the conjecture for 4 -regular graphs. In this paper, by applying the Combinatorial Nullstellensatz, we verify the conjecture for all graphs with maximum degree 4.

Plane Graphs with Maximum Degree Δ≥8 Are Entirely (Δ+3)-Colorable

Let be a plane graph with the sets of vertices, edges, and faces V, E, and F, respectively. If one can color all elements in using k colors so that any two adjacent or incident elements receive distinct colors, then G is said to be entirely k-colorable. Kronk and Mitchem [Discrete Math 5 (1973) 253-260] conjectured that every plane graph with maximum degree Δ is entirely -colorable. This conjecture has now been settled in Wang and Zhu (J Combin Theory Ser B 101 (2011) 490–501), where the authors asked: is every simple plane graph entirely -colorable? In this article, we prove that every simple plane graph with is entirely -colorable, and conjecture that every simple plane graph, except the tetrahedron, is entirely -colorable.

Neighbor sum distinguishing index of 2-degenerate graphs

We consider proper edge colorings of a graph G using colors in

Supereulerian graphs with small matching number and 2-connected hamiltonian claw-free graphs

{1,ldots ,k}

The Δ2-conjecture for L(2,1)-labelings is true for total graphs

{1,…,k}. Such a coloring is called neighbor sum distinguishing if for each pair of adjacent vertices u and v, the sum of the colors of the edges incident with u is different from the sum of the colors of the edges incident with v. The smallest value of k in such a coloring of G is denoted by

Strong List Edge Coloring of Subcubic Graphs

{mathrm ndi}_{Sigma }(G)