Xiangwen Li

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χg(G). In this note we will prove the following results. (1) Let H1 and H2 be two subgraphs of G such that V(H1)∩V(H2)=∅ and V(H1)∪V(H2)=V(G). Then χg(G)≤min{max{χg(H1), maxv∈V(H2)deg(v,G)+1},max{χg(H2), maxu∈V(H1)deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K3,3-minor, then χg(G)≤5.

A relaxation of the Bordeaux Conjecture

A ( c 1 , c 2 , ? , c k ) -coloring of a graph G is a mapping ? : V ( G ) ? { 1 , 2 , ? , k } such that for every i , 1 ? i ? k , G V i ] has maximum degree at most c i , where G V i ] denotes the subgraph induced by the vertices colored i . Borodin and Raspaud conjecture that every planar graph with neither 5 -cycles nor intersecting triangles is 3 -colorable. We prove in this paper that every planar graph with neither 5 -cycles nor intersecting triangles is (2, 0, 0)-colorable.

Planar graphs without 5-cycles and intersecting triangles are ( 1 , 1 , 0 ) -colorable

A ( c 1 , c 2 , ? , c k ) -coloring of G is a mapping ? : V ( G ) ? { 1 , 2 , ? , k } such that for every i , 1 ? i ? k , G V i has maximum degree at most c i , where G V i denotes the subgraph induced by the vertices colored i . Borodin and Raspaud conjecture that every planar graph without 5-cycles and intersecting triangles is ( 0 , 0 , 0 ) -colorable. We prove in this paper that such graphs are ( 1 , 1 , 0 ) -colorable.

On 3-choosable planar graphs of girth at least 4

Let G be a plane graph of girth at least 4. Two cycles of G are intersecting if they have at least one vertex in common. In this paper, we show that if a plane graph G has neither intersecting 4-cycles nor a 5-cycle intersecting with any 4-cycle, then G is 3-choosable, which extends one of Thomassens results [C. Thomassen, 3-list-coloring planar graphs of girth 5, J. Combin. Theory Ser. B 64 (1995) 101-107].

Hamilton cycles in 1-tough triangle-free graphs☆

A graph G is called triangle-free if G has no induced K3 as a subgraph. We set σ3=min{∑i=13d(vi)|{v1,v2,v3} is an independent set of vertices in G}. In this paper, we show that if G is a 1-tough and triangle-free graph of order n with n⩽σ3, then G is hamiltonian.

Nowhere-zero 3-flows of claw-free graphs

Let A denote an abelian group and G be a graph. If a graph G ? is obtained by repeatedly contracting nontrivial A -connected subgraphs of G until no such a subgraph left, we say G can be A -reduced to G ? . A graph is claw-free if it has no induced subgraph K 1 , 3 . Let N 1 , 1 , 0 denote the graph obtained from a triangle by adding two edges at two distinct vertices of the triangle, respectively. In this paper, we prove that if G is a simple 2-connected { claw , N 1 , 1 , 0 } -free graph, then G does not admit nowhere-zero 3-flow if and only if G can be Z 3 -reduced to two families of well characterized graphs or G is one of the five specified graphs.

Nowhere-zero 3-flows and Z 3 -connectivity in bipartite graphs

Abstract Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. Let F 12 be a family of graphs such that G ∈ F 12 if and only if G is a simple bipartite graph on 12 vertices and δ ( G ) = 4 . Let G be a simple bipartite graph on n vertices. It is proved in this paper that if δ ( G ) ≥ ⌈ n 4 ⌉ + 1 , then G admits a nowhere-zero 3-flow with only one exceptional graph. Moreover, if G ∉ F 12 with the minimum degree at least ⌈ n 4 ⌉ + 1 is Z 3 -connected. The bound is best possible in the sense that the lower bound for the minimum degree cannot be decreased.

A relaxation of the strong Bordeaux Conjecture

Let

Z 3 -Connectivity with Independent Number 2

c_1, c_2, \cdots, c_k

Group Chromatic Number of Halin Graphs

be