Guizhen Liu

List edge and list total colorings of planar graphs without 4-cycles

Let G be a planar graph with maximum degree Δ such that G has no cycle of length from 4 to k, where k ≥ 4. Then the list chromatic index χ′1(G) = Δ and the list total chromatic number χ″1(G) = Δ + 1 if (Δ, k) ∈ {(7, 4), (6, 5), (5, 8)}. Furthermore, χ′1(G) = Δ if (Δ, k) ∈ {(4, 14)}.

On connectivities of tree graphs

Let T(G) be the tree graph of a graph G with cycle rank r. Then κ(T(G)) ⩾ m(G) − r, where κ(T(G)) and m(G) denote the connectivity of T(G) and the length of a minimum cycle basis for G, respectively. Moreover, the lower bound of m(G) − r is best possible.

Some sufficient conditions for a graph to be of Cf 1

Abstract An f -coloring of a graph G is a coloring of edges of E ( G ) such that each color appears at each vertex v ∈ V ( G ) at most f ( v ) times. The minimum number of colors needed to f -color G is called the f -chromatic index χ f ′ ( G ) of G . Any graph G has f -chromatic index equal to Δ f ( G ) or Δ f ( G ) + 1 , where Δ f ( G ) = max v ∈ V { ⌈ d ( v ) f ( v ) ⌉ } . If χ f ′ ( G ) = Δ f ( G ) , then G is of C f 1; otherwise G is of C f 2. Some sufficient conditions for a graph to be of C f 1 are given.

Orthogonal (g, f)-factorizations in networks

Let G = (V, E) be a graph and let g and f be two integer-valued functions defined on V such that k ≤ g(x) ≤ f(x) for all x E V. Let H 1 , H 2 H k be subgraphs of G such that |E(H i )| = m, 1 ≤ i ≤ k, and V(H i ) ∩ V(H j ) = O when i ¬= j. In this paper, it is proved that every (mg + m - 1, mf - m + 1)-graph G has a (g, f)-factorization orthogonal to H i for i = 1, 2, k and shown that there are polynomial-time algorithms to find the desired (g, f)-factorizations.

Orthogonal ( g,f )-factorizations in graphs

Abstract Let G be a graph and let F = F 1 , F 2 , …, F m and H be a factorization and a subgraph of G, respectively. If H has exactly one edge in common with F i for all i, 1 ⩽ i ⩽ m , then we say that F is orthogonal to H . Let g and f be two integer-valued functions defined on V ( G ) such that g ( x ) ⩽ f ( x ) for every x ϵ V ( G ). In this paper it is proved that for any m -matching M of an ( mg + m − 1, mf − m + 1)-graph G, there exists a ( g , f )-factorization of G orthogonal to M .

The classification of complete graphsK n onf-coloring

AbstractAnf-coloring of a graphG=(V, E) is a coloring of edge setE such that each color appears at each vertexv ∈ V at mostf(v) times. The minimum number of colors needed tof-colorG is called thef-chromatic index χ′f(G) ofG. Any graphG hasf-chromatic index equal to Δf(G) or Δf(G) + 1 where

The structure of plane graphs with independent crossings and its applications to coloring problems

\Delta _f (G) = \mathop {\max }\limits_{v \in V} \left\{ {\left\lceil {\frac{{d(v)}}{{f(v)}}} \right\rceil } \right\}

Some problems on factorizations with constraints in bipartite graphs

. If χ′f(G) = Δf(G), thenG is ofCf 1; otherwiseG is ofCf 2. In this paper, the classification problem of complete graphs onf-coloring is solved completely.