The Entire Coloring of Series-Parallel Graphs

The entire chromatic number X_(vef)(G) of a plane graph G is the minimal number of colors needed for coloring vertices, edges and faces of G such that no two adjacent or incident elements are of the same color. Let G be a series-parallel plane graph, that is, a plane graph which contains no subgraphs homeomorphic to K_(4-) It is proved in this paper that X_(vef)(G)≤max{8, △(G) 2} and X_(vef)(G)=△ 1 if G is 2-connected and △(G)≥6.     

双圈图的零度与其匹配数之间的关系

Let G be a graph with n(G) vertices and m(G) be its matching number.The nullity of G,denoted by η(G),is the multiplicity of the eigenvalue zero of adjacency matrix of G.It is well known that if G is a tree,then η(G) = n(G)-2m(G).Guo et al.[Jiming GUO,Weigen YAN,Yeongnan YEH.On the nullity and the matching number of unicyclic graphs.Linear Alg.Appl.,2009,431:1293 1301]proved that if G is a unicyclic graph,then η(G)equals n(G)-2m(G)-1,n(G)-2m(G),or n(G)-2m(G) +2.In this paper,we prove that if G is a bicyclic graph,then η(G) equals n(G)-2m(G),n(G)-2m(G)±1,n(G)-2m(G)±2or n(G)-2m(G) + 4.We also give a characterization of these six types of bicyclic graphs corresponding to each nullity.     

Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a(G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g(G), and let 1≤r≤2Δ be an integer. In this paper, it is shown that there exists a constant c > 0 such that if g(G)≥cΔ r log(Δ2/r) then χa(G)≤Δ + r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that χ’a(G) = Δ if Δ≥4 and g(G)≥4; or Δ≥3 and g(G)≥5.     

Extremal Problems on Components and Loops in Graphs

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that Ω(G) ≤-4 are shown to be disconnected, and if Ω(G) ≥-2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G) =-2, then certainly the graph should be a tree.Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤-4, the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs.We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.     

INDEPENDENT-SET-DELETABLE FACTOR-CRITICAL POWER GRAPHS

It is said that a graph G is independent-set-deletable factor-critical (in short, ID-factor-critical), if, for every independent set 7 which has the same parity as |V(G)|, G-I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The k-th power of G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G3 and T(G) (the total graph of G) are ID-factor-critical, and G4 (when |V(G)| is even) is strongly IM-extendable; if G is 2-connected, then D2 is ID-factor-critical.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

The distance coloring of graphs

Let G be a connected graph with maximum degree Δ≥ 3.We investigate the upper bound for the chromatic number χγ(G) of the power graph Gγ.It was proved that χγ(G) ≤Δ(Δ-1)γ-1Δ-2+ 1 =:M + 1,where the equality holds if and only if G is a Moore graph.If G is not a Moore graph,and G satisfies one of the following conditions:(1) G is non-regular,(2) the girth g(G) ≤ 2γ- 1,(3)g(G) ≥ 2γ + 2,and the connectivity κ(G) ≥ 3 if γ≥ 3,κ(G) ≥ 4 but g(G) 6 if γ = 2,(4) Δis sufficiently larger than a given number only depending on γ,then χγ(G) ≤ M- 1.By means of the spectral radius λ1(G) of the adjacency matrix of G,it was shown that χ2(G) ≤λ1(G)2+ 1,where the equality holds if and only if G is a star or a Moore graph with diameter 2 and girth 5,and χγ(G)λ1(G)γ+1 ifγ≥3.     

Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs

A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex.A proper total-k-coloring of G is called neighbor sum distinguishing if∑_c(u)≠∑_c(v)for each edge uv∈E(G),where∑_c(v)denote the sum of the color of a vertex v and the colors of edges incident with v.The least number k needed for such a total coloring of G,denoted byχ∑"is the neighbor sum distinguishing total chromatic number.Pilsniak and Wozniak conjecturedχ∑"(G)≤Δ(G)+3 for any simple graph with maximum degreeΔ(G).By using the famous Combinatorial Nullstellensatz,we prove that above conjecture holds for any triangle free IC-planar graph with△(G)≥7.Moreover,it holds for any triangle free planar graph withΔ(G)≥6.     

关于H-cordial图和semi-H-cordial图的若干结论（英文）

I. Cahit calls a graph H-cordial if it is possible to label the edges with the numbers from the set{1,-1} in such a way that, for some k, at each vertex v the sum of the labels on the edges incident with v is either k or-k and the inequalities |v(k)-v(-k)| ≤ 1 and|e(1)-e(-1)| ≤ 1 are also satisfied. A graph G is called to be semi-H-cordial, if there exists a labeling f, such that for each vertex v, |f(v)| ≤ 1, and the inequalities |e_f(1)-e_f(-1)| ≤ 1 and |vf(1)-vf(-1)| ≤ 1 are also satisfied. An odd-degree(even-degree) graph is a graph that all of the vertex is odd(even) vertex. Three conclusions were proved:(1) An H-cordial graph G is either odd-degree graph or even-degree graph;(2) If G is an odd-degree graph, then G is H-cordial if and only if |E(G)| is even;(3) A graph G is semi-H-cordial if and only if |E(G)| is even and G has no Euler component with odd edges.     

A sufficient condition for planar graphs with maximum degree 8 to be 9-totally colorable

A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ'(G) is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph G has maximum degree Δ≥ 9, then χ'(G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ'(G) = 9.     

Injective coloring of planar graphs

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic numberχ_i(G) of a graph G is the least k such that there is an injective k-coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree Δ, then (1) χ_i(G)=Δ if either g≥20 and Δ≥3, or g≥7 and Δ≥71; (2) χ_i(G)≤Δ+1 if g≥11; (3) χ_i(G)≤Δ+2 if g≥8.     

The total chromatic number of Pseudo-Halin graphs with lower degree

The total chromatic number χ_T(G) of a graph G is the least number of colors needed to color the vertices and the edges of G such that no adjacent or incident elements receive the same color. The Total Coloring Conjecture(TCC) states that for every simple graph G, χ_T(G)≤Δ(G)+2. In this paper, we show that χ_T(G)=Δ(G)+1 for all pseudo-Halin graphs with Δ(G)=4 and 5.     

Total chromatic number of one kind of join graphs

The total chromatic number χ_T(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if χ_T(G)=Δ(G)+1. In this paper we prove that the join of a complete inequibipartite graph K_n1,n2 and a path P_m is of Type 1.     

Circular game chromatic number of graphs

In a circular r-colouring game on G, Alice and Bob take turns colouring the vertices of G with colours from the circle S(r) of perimeter r. Colours assigned to adjacent vertices need to have distance at least 1 in S(r). Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χ_cg(G) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r-colouring game on G. This paper proves that for any graph G, , where is the game colouring number of G. This upper bound is shown to be sharp for forests. It is also shown that for any graph G, χ_cg(G)≤2χ_a(G)(χ_a(G)+1), where χ_a(G) is the acyclic chromatic number of G. We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles.     

Improvement on Brooks' chromatic bound for a class of graphs

Brooks' Theorem says that if for a graph G,Δ(G)=n, then G is n-colourable, unless (1) n=2 and G has an odd cycle as a component, or (2) n>2 and K_n+1 is a component of G. In this paper we prove that if a graph G has none of some three graphs (K_1,3;K₅?e and H) as an induced subgraph and if Δ(G)?6 and d(G)<Δ(G), then χ(G)<Δ(G). Also we give examples to show that the hypothesis Δ(G)?6 can not be non-trivially relaxed and the graph K₅?e can not be removed from the hypothesis. Moreover, for a graph G with none of K_1,3;K₅?e and H as an induced subgraph, we verify Borodin and Kostochka's conjecture that if for a graph G,Δ(G)?9 and d(G)<Δ(G), then χ(G)<Δ(G).     

On the b-continuity property of graphs

This paper deals with b-colorings of a graph G, that is, proper colorings in which for each color c, there exists at least one vertex colored by c such that its neighbors are colored by each other color. The b-chromatic numberb(G) of a graph G is the maximum number of colors for which G has a b-coloring. It is easy to see that every graph G has a b-coloring using χ(G) colors.We say that G is b-continuous iff for each k, χ(G)?k?b(G), there exists a b-coloring with k colors. It is well known that not all graphs are b-continuous. We call b-spectrumS_b(G) of G to be the set of integers k for which there is a b-coloring of G by k colors. We show that for any finite integer set I, there exists a graph whose b-spectrum is I and we investigate the complexity of the problem of deciding whether a graph G is b-continuous, even if b-colorings using χ(G) and b(G) colors are given.     

Edge covering coloring of nearly bipartite graphs

LetG be a simple graph with vertex setV(G) and edge setE(G). A subsetS ofE(G) is called an edge cover ofG if the subgraph induced byS is a spanning subgraph ofG. The maximum number of edge covers which form a partition ofE(G) is called edge covering chromatic number ofG, denoted by χ′_c(G). It known that for any graphG with minimum degreeδ,δ -1 ≤χ′_c(G) ≤δ. If χ′_c(G) =δ, thenG is called a graph of CI class, otherwiseG is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification of nearly bipartite graph and give some sufficient conditions for a nearly bipartite graph to be of CI class.     

ENTIRE CHROMATIC NUMBER AND △-MATCHING OF OUTERPLANE GRAPHS

Let G be an outerplane graph with maximum degree A and the entire chromatic number Xvef（G）. This paper proves that if △ ≥6, then △＋ 1≤Xvef（G）≤△＋ 2, and Xvef （G） = △＋ 1 if and only if G has a matching M consisting of some inner edges which covers all its vertices of maximum degree.     

List coloring of graphs having cycles of length divisible by a given number

Let G be a graph and χ_l(G) denote the list chromatic number of G. In this paper we prove that for every graph G for which the length of each cycle is divisible by l (l≥3), χ_l(G)≤3.