图的边覆盖染色中的分类问题(英文)

设 G是一个图 ,其边集是 E( G) ,E( G)的一个子集 S称为 G的一个边覆盖 ,若 G的每一点都是 S中一条边的端点 .G的一个 (正常 )边覆盖染色是对 G的边进行染色 ,使得每一色组都是 G的一个边覆盖 ,使 G有 (正常 )边覆盖染色所需最多颜色数 ,称为 G的边覆盖色数 ,用χ′c( G)表示 .已知的结果是对于任意简单图 G,都有 δ- 1≤ χ′c( G)≤ δ,δ是 G的最小度 .若 χ′c( G) =δ,则称 G是 CI类的 ;否则称为 CII类的 .本文主要研究了平面图及平衡的完全 r分图的分类问题     

The classification of complete graphsK n onf-coloring

Anf-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 Δ _f (G) or Δ _f (G) + 1 where $\Delta _f (G) = \mathop {\max }\limits_{v \in V} \left\{ {\left\lceil {\frac{{d(v)}}{{f(v)}}} \right\rceil } \right\}$ . If χ′ _f (G) = Δ _f (G), thenG is ofC _f 1; otherwiseG is ofC _f 2. In this paper, the classification problem of complete graphs onf-coloring is solved completely.     

On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

Circular chromatic index of graphs of maximum degree 3

This paper proves that if G is a graph (parallel edges allowed) of maximum degree 3, then χ′_c(G) ≤ 11/3 provided that G does not contain H₁ or H₂ as a subgraph, where H₁ and H₂ are obtained by subdividing one edge of K (the graph with three parallel edges between two vertices) and K₄, respectively. As χ′_c(H₁) = χ′_c(H₂) = 4, our result implies that there is no graph G with 11/3 < χ′_c(G) < 4. It also implies that if G is a 2‐edge connected cubic graph, then χ′_c(G) ≤ 11/3. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 325–335, 2005     

The bipartite edge frustration of composite graphs

The smallest number of edges that have to be deleted from a graph to obtain a bipartite spanning subgraph is called the bipartite edge frustration of G and denoted by φ(G). In this paper we determine the bipartite edge frustration of some classes of composite graphs.     

Collapsible biclaw-free graphs

A graph is called biclaw-free if it has no biclaw as an induced subgraph. In this note, we prove that if G is a connected bipartite biclaw-free graph with δ(G)?5, then G is collapsible, and of course supereulerian. This bound is best possible.     

Proper connection number of bipartite graphs

An edge-colored graph G is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph G, denoted by pc(G), is the smallest number of colors that are needed to color the edges of G in order to make it proper connected. In this paper, we obtain the sharp upper bound for pc(G) of a general bipartite graph G and a series of extremal graphs. Additionally, we give a proper 2-coloring for a connected bipartite graph G having δ(G) ≥ 2 and a dominating cycle or a dominating complete bipartite subgraph, which implies pc(G) = 2. Furthermore, we get that the proper connection number of connected bipartite graphs with δ ≥ 2 and diam(G) ≤ 4 is two.     

Some properties onf-edge covered critical graphs

LetG(V, E) be a simple graph, and letf be an integer function onV with 1 ≤f(v) ≤d(v) to each vertexv ∈V. An f-edge cover-coloring of a graphG is a coloring of edge setE such that each color appears at each vertexv ∈V at leastf(v) times. Thef-edge cover chromatic index ofG, denoted by χ′ _fc (G), is the maximum number of colors such that anf-edge cover-coloring ofG exists. Any simple graphG has anf-edge cover chromatic index equal to δ_f or δ _f - 1, where $\delta _f = \mathop {\min }\limits_{\upsilon \in V} \{ \left\lfloor {\frac{{d(v)}}{{f(v)}}} \right\rfloor \} $ . LetG be a connected and not complete graph with χ′ _fc (G)=δ _f-1, if for eachu, v ∈V and e =uv ?E, we have ÷ _fc (G + e) > ÷ _fc (G), thenG is called anf-edge covered critical graph. In this paper, some properties onf-edge covered critical graph are discussed. It is proved that ifG is anf-edge covered critical graph, then for eachu, v ∈V and e =uv ?E there existsw ∈ {u, v } withd(w) ≤ δ _f (f(w) + 1) - 2 such thatw is adjacent to at leastd(w) - δ _f + 1 vertices which are all δ _f -vertex inG.     

Adjacent strong edge coloring of graphs

For a graph G(V, E), if a proper k-edge coloring ƒ is satisfied with C(u) ≠ C(v) for uv ∈ E(G), where $\text{C(u) = {ƒ(uv) | uv ∈ E}}$, then ƒ is called k-adjacent strong edge coloring of G, is abbreviated k-ASEC, and χ′_as(G) = min{k | k-ASEC of G} is called the adjacent strong edge chromatic number of G. In this paper, we discuss some properties of χ′_as(G), and obtain the χ′_as(G) of some special graphs and present a conjecture: if G are graphs whose order of each component is at least six, then χ′_as(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree of G.     

Odd cycles and matrices with integrality properties

A cycle of a bipartite graphG(V⁺, V^?; E) is odd if its length is 2 (mod 4), even otherwise. An odd cycleC is node minimal if there is no odd cycleC′ of cardinality less than that ofC′ such that one of the following holds:C′ ∩V ⁺ ?C ∩V ⁺ orC′ ∩V ^? ?C ∩V ^?. In this paper we prove the following theorem for bipartite graphs: For a bipartite graphG, one of the following alternatives holds:

-All the cycles ofG are even.

-G has an odd chordless cycle.

-For every node minimal odd cycleC, there exist four nodes inC inducing a cycle of length four.

-An edge (u, v) ofG has the property that the removal ofu, v and their adjacent nodes disconnects the graphG.

To every (0, 1) matrixA we can associate a bipartite graphG(V⁺, V^?; E), whereV ⁺ andV ^? represent respectively the row set and the column set ofA and an edge (i,j) belongs toE if and only ifa _ij = 1. The above theorem, applied to the graphG(V⁺, V^?; E) can be used to show several properties of some classes of balanced and perfect matrices. In particular it implies a decomposition theorem for balanced matrices containing a node minimal odd cycleC, having the property that no four nodes ofC induce a cycle of length 4. The above theorem also yields a proof of the validity of the Strong Perfect Graph Conjecture for graphs that do not containK ₄?e as an induced subgraph.     

Convex hull of the edges of a graph and near bipartite graphs

First we characterize the convex hull of the edges of a graph, edges viewed as the characteristic function of the hereditary closure of some subset of the 2-elements set of a finite set X. This characterization becomes more simple for a class of graphs that we call near bipartite, NBP in short. This class is then characterized as the class of graphs such that ?x?X, G_X\r(x), the induced subgraph of the complementary of the neighbourhood of x, is bipartite. We made a partial study of this class, whose interest is justified by the constatation that the following classes are strictly include: $\text{L(G)}$ the edge complementary of the line graph of G. NBP, $\text{K}{}_{13}$-free graphs.     

On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ _r (G) ≤ μ(G), where μ _r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ _r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.     

Matchings in 3-vertex-critical graphs: The odd case

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G-v)<γ(G), for every vertex v in G. A graph G is said to be factor-critical if G-v has a perfect matching for every choice of v∈V(G).In this paper, we present two main results about 3-vertex-critical graphs of odd order. First we show that any such graph with positive minimum degree and at least 11 vertices which has no induced subgraph isomorphic to the bipartite graph K_1,5 must contain a near-perfect matching. Secondly, we show that any such graph with minimum degree at least three which has no induced subgraph isomorphic to the bipartite graph K_1,4 must be factor-critical. We then show that these results are best possible in several senses and close with a conjecture.     

(g, f)-factorizations orthogonal to a subgraph in graphs

LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)⩽(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively. Project supported by the National Natural Science Foundation of China.     

A note on acyclic domination number in graphs of diameter two

A subset S of the vertex set of a graph G is called acyclic if the subgraph it induces in G contains no cycles. S is called an acyclic dominating set of G if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by γ_a(G), is called the acyclic domination number of G. Hedetniemi et al. [Acyclic domination, Discrete Math. 222 (2000) 151-165] introduced the concept of acyclic domination and posed the following open problem: if δ(G) is the minimum degree of G, is γ_a(G)?δ(G) for any graph whose diameter is two? In this paper, we provide a negative answer to this question by showing that for any positive k, there is a graph G with diameter two such that γ_a(G)-δ(G)?k.     

On a Class of P5-Free Graphs

A graph G is perfect in the sense of Berge if for every induced subgraph G′ of G, the chromatic number χ(G′) equals the largest number ω(G′) of pairwise adjacent vertices in G′. The Strong Perfect Graph Conjecture asserts that a graph G is perfect if, and only if, neither G nor its complement ? contains an odd chordless cycle of length at least five. We prove that the conjecture is true for a class of P₅-free graphs.     

Graphoidal bipartite graphs

A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths inG such that every path in ψ has at least two vertices, every vertex ofG is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. Let Ω (ψ) denote the intersection graph of ψ. A graph G is said to be graphoidal if there exists a graphH and a graphoidal cover ψof H such that G is isomorphic to Ω(ψ). In this paper we study the properties of graphoidal graphs and obtain a forbidden subgraph characterisation of bipartite graphoidal graphs.     

Graphs Having (2 mod d)-Cycles

It is known that if G is a graph with minimum degree δ at least d+1, then G has a cycle of length 2 mod d. We show that if G is also bipartite, then G has a cycle of length 2 mod 2d. Both these bounds are tight in terms of minimum degree. However, we show that if G is a graph with δ ≥ d and had neither K_d nor K_d,d as an induced subgraph, then G has a cycle of length 2 mod d. If G is also bipartite, then G has a cycle of length 2 mod 2d. If G is a 2-connected graph with δ ≥ d and is not congruent to K_d nor K_d,d' (for d' ≥ d) then G has a cycle of length 2 mod d. If G is also bipartite, then G has a cycle of length 2 mod 2d.     

Covering of graphs by complete bipartite subgraphs; Complexity of 0–1 matrices

We prove that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log₂ n]+1 complete bipartite subgraphs ofG. If the weight of a subgraph is the number of its vertices, then there always exists a cover with total weightc(n ²/logn) and this bound is sharp apart from a constant factor. Our result answers a problem of T. G. Tarján. Dedicated to Paul Erdős on his seventieth birthday     

Some Asymptotic ith Ramsey numbers

Let G be a graph with chromatic number χ(G) and let t(G) be the minimum number of vertices in any color class among all χ(G)-vertex colorings of G. Let H′ be a connected graph and let H be a graph obtained by subdividing (adding extra vertices to) a fixed edge of H′. It is proved that if the order of H is sufficiently large, the ith Ramsey number r_i(G, H) equals $\text{[}\text{((χ(G)?1)(|}\text{H}\text{|?1)+t(G)?1)}\text{i}\text{]+1}$.