Enqiang Zhu

On dominating sets of maximal outerplanar and planar graphs

A set D ? V ( G ) of a graph G is a dominating set if every vertex v ? V ( G ) is either in D or adjacent to a vertex in D . The domination number γ ( G ) of a graph G is the minimum cardinality of a dominating set of G . Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n -vertex maximal outerplanar graph having t vertices of degree 2, then γ ( G ) ? n + t 4 . We improve their upper bound by showing γ ( G ) ? n + k 4 , where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ ( G ) ? 5 n 16 for a Hamiltonian maximal planar graph G with n ? 7 vertices.

A note on local coloring of graphs

A local k-coloring of a graph G is a function f : V ( G ) ? { 1 , 2 , ? , k } such that for each S ? V ( G ) , 2 ? | S | ? 3 , there exist u , v ? S with | f ( u ) - f ( v ) | at least the size of the subgraph induced by S. The local chromatic number of G is ? ? ( G ) = min ? { k : G has a local k -coloring } . Chartrand et al. 2 asked: does there exist a graph G k such that ? ? ( G k ) = ? ( G k ) = k ? Furthermore, they conjectured that for every positive integer k, there exists a graph G k with ? ? ( G ) = k such that every local k-coloring of G k uses all of the colors 1 , 2 , ? , k . In this paper we give a affirmative answer to the problem and confirm the conjecture. We study the local k-coloring of graphs.We prove that there exists a graph G k such that ? ? ( G k ) = ? ( G k ) = k for any k. This result gives a affirmative answer to a problem proposed by Chartrand et al. 2.We prove that there exists a graph G k with ? ? ( G ) = k such that every local k-coloring of G k uses all of the colors 1 , 2 , ? , k . This result confirms a conjecture proposed by Chartrand et al. 2.

Total domination and open packing in some chemical graphs

In this paper we obtain some bounds and exact values of the total domination numbers and open packing numbers for pyrene torus, hexabenzocoronene, H-phenylenic nanotube and H-napthelenic nanotube.

NP-completeness of local colorings of graphs

We study the local k-coloring of graphs.We prove that it is NP-complete to determine whether a graph has a local k-coloring for fixed k=4 or k=2t1, where t3.We prove that it is NP-complete to determine whether a planar graph has a local 5-coloring even restricted to the maximum degree =7or8. A local k-coloring of a graph G is a function f:V(G){1,2,,k} such that for each SV(G), 2|S|3, there exist u,vS with |f(u)f(v)| at least the size of the subgraph induced by S. The local chromatic number of G is (G)=min{k:Ghas a localk-coloring}. In this paper, we show that it is NP-complete to determine whether a graph has a local k-coloring for fixed k=4 or k=2t1, where t3. In particular, it is NP-complete to determine whether a planar graph has a local 5-coloring even restricted to the maximum degree =7or8.

Semitotal Domination in Claw-Free Cubic Graphs

The semitotal domination number of a graph G without isolated vertices is the minimum cardinality of a set S of vertices of G such that every vertex in

Acyclically 4-colorable triangulations

V(G){\setminus } S

Construction of acyclically 4-colourable planar triangulations with minimum degree 4

V(G)\S is adjacent to at least one vertex in S, and every vertex in S is within distance 2 of another vertex of S. In Henning and Marcon (Ann Comb 20(4):1–15, 2016), it was shown that every connected claw-free cubic graph G of order n has semitotal domination number at most