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1.
Let G be a simple graph with no isolated edge. An Ⅰ-total coloring of a graph G is a mapping φ : V(G) ∪ E(G) → {1, 2, ···, k} such that no adjacent vertices receive the same color and no adjacent edges receive the same color. An Ⅰ-total coloring of a graph G is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, we have C_φ(u) = C_φ(v), where C_φ(u) denotes the set of colors of u and its incident edges. The minimum number of colors required for an adjacent vertex distinguishing Ⅰ-total coloring of G is called the adjacent vertex distinguishing Ⅰ-total chromatic number, denoted by χ_at~i(G).In this paper, we characterize the adjacent vertex distinguishing Ⅰ-total chromatic number of outerplanar graphs.  相似文献   

2.
For a proper edge coloring c of a graph G,if the sets of colors of adjacent vertices are distinct,the edge coloring c is called an adjacent strong edge coloring of G.Let c i be the number of edges colored by i.If |c i c j | ≤ 1 for any two colors i and j,then c is an equitable edge coloring of G.The coloring c is an equitable adjacent strong edge coloring of G if it is both adjacent strong edge coloring and equitable edge coloring.The least number of colors of such a coloring c is called the equitable adjacent strong chromatic index of G.In this paper,we determine the equitable adjacent strong chromatic index of the joins of paths and cycles.Precisely,we show that the equitable adjacent strong chromatic index of the joins of paths and cycles is equal to the maximum degree plus one or two.  相似文献   

3.
《数学季刊》2016,(2):147-154
Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) 6= C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt(G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K8,n are discussed in this paper. Particularly, the VDIET chromatic number of K8,n are obtained.  相似文献   

4.
Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)~(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.  相似文献   

5.
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.  相似文献   

6.
D(β)-vertex-distinguishing total coloring of graphs   总被引:2,自引:0,他引:2  
A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger thanβhave different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.  相似文献   

7.
A linear coloring of a graph G is a proper vertex coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths.The linear chromatic number lc(G) o...  相似文献   

8.
Zhang  Xin  Niu  Bei  Li  Yan  Li  Bi 《数学学报(英文版)》2021,37(8):1293-1302
The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-k-coloring of a graph is a vertex coloring using k distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every d-degenerate graph with maximum degree at most Δ is equitably tree-k-colorable for every integer k ≥(Δ + 1)/2 provided that Δ≥ 9.818 d,confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.  相似文献   

9.
An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.  相似文献   

10.
Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C(x) denote the set of colors of vertex x and of the edges incident with x, we call C(x) the color set of x. If C(u)≠ C(v) for any two different vertices u and v of V(G), then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by χ_(vt)~e(G) and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K_(7,n)(7 ≤ n ≤ 95) is discussed in this paper and the VDET chromatic number of K_(7,n)(7 ≤ n ≤ 95) has been obtained.  相似文献   

11.
A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uvE(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by x Aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.  相似文献   

12.
A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by ???? s (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with ??(G) ?? 4, respectively.  相似文献   

13.
A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ aa (G) ≤ 32Δ. Supported by the Natural Science Foundation of Gansu Province (3ZS051-A25-025)  相似文献   

14.
The minimum number of total independent partition sets of VE of graph G(V,E) is called the total chromatic number of G denoted by χ t (G). If the difference of the numbers of any two total independent partition sets of VE is no more than one, then the minimum number of total independent partition sets of VE is called the equitable total chromatic number of G, denoted by χ et (G). In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order. Supported by the National Natural Science Foundation of China (No. 10771091).  相似文献   

15.
An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ=(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ=*(G). In this paper, we investigate the asymptotic behavior of these coloring parameters in the probability space G(n,p) of random graphs. We prove that if n?1/5+? < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ=(G(n,p)) = (1 + o(1))χ(G(n,p)) holds (where χ(G(n,p)) is the ordinary chromatic number of G(n,p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n,p)) ≤ χ=(G(n,p)) ≤ (2 + o(1))χ(G(n,p)). Concerning the equitable chromatic threshold, we prove that if n?(1??) < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ=* (G(n,p)) ≤ (2 + o(1))χ(G(n,p)) holds, and if < p < 0.99 for some 0 < ?, then almost surely we have χ(G(n,p)) ≤ χ=*(G(n,p)) = O?(χ(G(n,p))). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009  相似文献   

16.
For a nontrivial connected graph G, let c: V (G) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) ≠ NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number x s (G). A study is made of the set chromatic number of the join G+H of two graphs G and H. Sharp lower and upper bounds are established for x s (G + H) in terms of x s (G), x s (H), and the clique numbers ω(G) and ω(H).  相似文献   

17.
本文研究了图的符号团边控制数的问题.利用鸽巢原理,获得了图KnPmKnCm的符号团边控制数,推广了已有的结果.  相似文献   

18.
A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.   相似文献   

19.
Coloring of the graph products, especially vertex and edge coloring, has been widely researched for all types of graph products. For total graph coloring, as combination of edge and vertex coloring, Behzad and Vizing set Total Coloring Conjecture in mid 1960s. In this paper, we prove the conjecture for two specific direct graph products, for direct product of path and arbitrary graph G, P n ×G, where χ′(G)=Δ(G), and expand the proof onto direct product of arbitrary cycle and a path P n , C m ×P n . At the same time, the proofs provide the algorithms to color such graphs.  相似文献   

20.
An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G. For a family F of graphs, the acyclic chromatic number of F, denoted by a(F), is defined as the maximum a(G) over all the graphs GF. In this paper we show that a(F)=8 where F is the family of graphs of maximum degree 5 and give a linear time algorithm to achieve this bound.  相似文献   

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