A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In...A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In 2015,Pilśniak and Wozniak conjectured that every graph with maximum degreeΔhas an NSD total(Δ+3)-coloring.Recently,Yang et al.proved that the conjecture holds for planar graphs withΔ≥10,and Qu et al.proved that the list version of the conjecture also holds for planar graphs withΔ≥13.In this paper,we improve their results and prove that the list version of the conjecture holds for planar graphs withΔ≥10.展开更多
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, we prove that every 1-planar graph G with maximum degree △（G） 〉 12 and girth at least five...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, we prove that every 1-planar graph G with maximum degree △（G） 〉 12 and girth at least five is totally （△（G）＋1）-colorable.展开更多
基金supported by the National Natural Science Foundation of China (No.12271438, No.12071370 and U1803263)the Science Found of Qinhai Province (No.2022-ZJ-753)+2 种基金Shaanxi Fundamental Science Research Project for Mathematics and Physics (No.22JSZ009)Shangluo University Doctoral Initiation Fund Project(No.22SKY112)Shangluo University Key Disciplines Project (Discipline name:Mathematics)。
文摘A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In 2015,Pilśniak and Wozniak conjectured that every graph with maximum degreeΔhas an NSD total(Δ+3)-coloring.Recently,Yang et al.proved that the conjecture holds for planar graphs withΔ≥10,and Qu et al.proved that the list version of the conjecture also holds for planar graphs withΔ≥13.In this paper,we improve their results and prove that the list version of the conjecture holds for planar graphs withΔ≥10.
基金Supported by NSFC(Nos.11771443,12001481,12071265)Future Scientists Program of China University of Mining and Technology(No.2023WLKXJ125)Natural Science Foundation of Shandong Province(No.ZR2021MA103)
基金supported by National Natural Science Foundation of China(Grant No.11271006)
文摘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, we prove that every 1-planar graph G with maximum degree △（G） 〉 12 and girth at least five is totally （△（G）＋1）-colorable.
