Let G be a simple graph. Let f be a mapping from V（G） U E（G） to {1, 2,..., k}. Let Cf（v） = {f（v）} U {f（vw）｜w ∈ V（G）,vw ∈ E（G）} for every v ∈ V（G）. If f is a k-propertotal-coloring, and if Cf（u） ...Let G be a simple graph. Let f be a mapping from V（G） U E（G） to {1, 2,..., k}. Let Cf（v） = {f（v）} U {f（vw）｜w ∈ V（G）,vw ∈ E（G）} for every v ∈ V（G）. If f is a k-propertotal-coloring, and if Cf（u） ≠ Cf（v） for uv ∈ V（G）,uv E E（G）, then f is called k-adjacentvertex-distinguishing total coloring of G（k-AVDTC of G for short）. Let χat（G） = min{k｜G has a k-adjacent-vertex-distinguishing total coloring}. Then χat（G） is called the adjacent-vertex-distinguishing total chromatic number. The adjacent-vertex-distinguishing total chromatic number on the Cartesion product of path Pm and complete graph Kn is obtained.展开更多
Let G be a simple graph. Let f be a mapping from V (G) ∪ E(G) to {1,2,...,k}. Let Cf(v) = {f(v)} ∪ {f(vw)|w ∈ V (G),vw ∈ E(G)} for every v ∈ V (G). If f is a k-proper- total-coloring, and for u,v ∈ V (G),uv ∈ E...Let G be a simple graph. Let f be a mapping from V (G) ∪ E(G) to {1,2,...,k}. Let Cf(v) = {f(v)} ∪ {f(vw)|w ∈ V (G),vw ∈ E(G)} for every v ∈ V (G). If f is a k-proper- total-coloring, and for u,v ∈ V (G),uv ∈ E(G), we have Cf(u) = Cf(v), then f is called a k- adjacent-vertex-distinguishing total coloring (k-AV DTC for short). Let χat(G) = min{k|G have a k-adjacent-vertex-distinguishing total coloring}. Then χat(G) is called the adjacent-vertex- distinguishing total chromatic number (AV DTC number for short)...展开更多
The total chromatic number xT（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 c...The total chromatic number xT（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 xT（G） =△（G）＋1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.展开更多
A proper total-coloring of graph G is said to be?equitable if the number of elements (vertices and edges) in any?two color classes differ by at most one, which the required?minimum number of colors is called the equit...A proper total-coloring of graph G is said to be?equitable if the number of elements (vertices and edges) in any?two color classes differ by at most one, which the required?minimum number of colors is called the equitable total chromatic?number. In this paper, we prove some theorems on equitable?total coloring and derive the equitable total chromatic numbers?of Pm V?Sn, Pm V?Fn and Pm V Wn.展开更多
A planar graph G is called a i pseudo outerplanar graph if there is a subset V 0V(G),|V 0|=i, such that G-V 0 is an outerplanar graph. In particular, when G-V 0 is a forest, G is called a i...A planar graph G is called a i pseudo outerplanar graph if there is a subset V 0V(G),|V 0|=i, such that G-V 0 is an outerplanar graph. In particular, when G-V 0 is a forest, G is called a i pseudo tree. In this paper, the following results are proved: (i) The conjecture on the total coloring is true for all 1 pseudo outerplanar graphs; (ii) χ t(G)=Δ(G)+1 for any 1 pseudo outerplanar graph G with Δ(G)6 and for any 1 pseudo tree G with Δ(G)3, where χ t(G) is the total chromatic number of a graph G .展开更多
An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees.The purpose of this paper is to derive exact values of acyclic chromatic n...An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees.The purpose of this paper is to derive exact values of acyclic chromatic number of some graphs.展开更多
Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） ...Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） and S（v） S（u） for any two adjacent vertices u and v, then f is called a Smarandachely adjacent vertex distinguishing proper edge col- oring using k colors, or k-SA-edge coloring. The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number, or SA- edge chromatic number for short, and denoted by Xsa（G）. In this paper, we have discussed the SA-edge chromatic number of K4 V Kn.展开更多
Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has...Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ? 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.展开更多
Let G = （V, E） be a simple connected graph, and ｜V（G）｜ ≥ 2. Let f be a mapping from V（G） ∪ E（G） to {1,2…, k}. If arbitary uv ∈ E（G）,f（u） ≠ f（v）,f（u） ≠ f（uv）,f（v） ≠ f（uv）; arbitary uv, uw...Let G = （V, E） be a simple connected graph, and ｜V（G）｜ ≥ 2. Let f be a mapping from V（G） ∪ E（G） to {1,2…, k}. If arbitary uv ∈ E（G）,f（u） ≠ f（v）,f（u） ≠ f（uv）,f（v） ≠ f（uv）; arbitary uv, uw ∈ E（G）（v ≠ w）, f（uv） ≠ f（uw）;arbitary uv ∈ E（G） and u ≠ v, C（u） ≠ C（v）, whereC（u）={f（u）}∪{f（uv）｜uv∈E（G）}.Then f is called a k-adjacent-vertex-distinguishing-proper-total coloring of the graph G（k-AVDTC of G for short）. The number min{k｜k-AVDTC of G} is called the adjacent vertex-distinguishing total chromatic number and denoted by χat（G）. In this paper we prove that if △（G） is at least a particular constant and δ ≥32√△ln△, then χat（G） ≤ △（G） ＋ 10^26 ＋ 2√△ln△.展开更多
A vertex distinguishing equitable total coloring of graph G is a proper total coloring of graph G such that any two distinct vertices＇ coloring sets are not identical and the difference of the elements colored by any...A vertex distinguishing equitable total coloring of graph G is a proper total coloring of graph G such that any two distinct vertices＇ coloring sets are not identical and the difference of the elements colored by any two colors is not more than 1. In this paper we shall give vertex distinguishing equitable total chromatic number of join graphs Pn VPn, Cn VCn and prove that they satisfy conjecture 3, namely, the chromatic numbers of vertex distinguishing total and vertex distinguishing equitable total are the same for join graphs Pn V Pn and Cn ∨ Cn.展开更多
Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct verti...Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct vertices u and v of V（G）, then f is called a vertex- distinguishing total k-coloring of G. The minimum number k for which there exists a vertex- distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by Xvt（G）. The vertex-disjoint union of two cycles of length n is denoted by 2Cn. We will obtain Xvt（2Cn） in this paper.展开更多
Let G be a simple graph and f be a proper total kcoloring of G. The color set of each vertex v of G is the set of colors appearing on v and the edges incident to v. The coloring f is said to be an adjacent vertex-dist...Let G be a simple graph and f be a proper total kcoloring of G. The color set of each vertex v of G is the set of colors appearing on v and the edges incident to v. The coloring f is said to be an adjacent vertex-distinguishing total coloring if the color sets of any two adjacent vertices are distinct. The minimum k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number of G. The join graph of two vertex-disjoint graphs is the graph union of these two graphs together with all the edges that connect the vertices of one graph with the vertices of the other. The adjacent vertex-distinguishing total chromatic numbers of the join graphs of an empty graph of order s and a complete graph of order t are determined.展开更多
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....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 Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.展开更多
The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regula...The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) ? 2 3 |V (G)|+ 23 6 , where d(G) denotes the degree of a vertex in G, then χT (G) ? d(G) + 2.展开更多
The total chromatic number XT(G) of graph G is the least number of colorsassigned to VE(G) such that no adjacent or incident elements receive the same color.Gived graphs G1,G2, the join of G1 and G2, denoted by G1∨G2...The total chromatic number XT(G) of graph G is the least number of colorsassigned to VE(G) such that no adjacent or incident elements receive the same color.Gived graphs G1,G2, the join of G1 and G2, denoted by G1∨G2, is a graph G, V(G) =V(GI)∪V(G2) and E(G) = E(G1)∪E(G2) ∪{uv | u∈(G1), v ∈ V(G2)}. In this paper, it's proved that if v(G) = v(H), both Gc and Hc contain perfect matching and one of the followings holds: (i)Δ(G) =Δ(H) and there exist edge e∈ E(G), e' E E(H)such that both G-e and H-e' are of Class l; (ii)Δ(G)<Δ(H) and there exixst an edge e ∈E(H) such that H-e is of Class 1, then, the total coloring conjecture is true for graph G ∨H.展开更多
Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoi...Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.展开更多
基金the Science and Research Project of Education Department of Gansu Province (0501-02)
文摘Let G be a simple graph. Let f be a mapping from V（G） U E（G） to {1, 2,..., k}. Let Cf（v） = {f（v）} U {f（vw）｜w ∈ V（G）,vw ∈ E（G）} for every v ∈ V（G）. If f is a k-propertotal-coloring, and if Cf（u） ≠ Cf（v） for uv ∈ V（G）,uv E E（G）, then f is called k-adjacentvertex-distinguishing total coloring of G（k-AVDTC of G for short）. Let χat（G） = min{k｜G has a k-adjacent-vertex-distinguishing total coloring}. Then χat（G） is called the adjacent-vertex-distinguishing total chromatic number. The adjacent-vertex-distinguishing total chromatic number on the Cartesion product of path Pm and complete graph Kn is obtained.
基金the National Natural Science Foundation of China (No.10771091)the Science and Research Project of the Education Department of Gansu Province (No.0501-02)
文摘Let G be a simple graph. Let f be a mapping from V (G) ∪ E(G) to {1,2,...,k}. Let Cf(v) = {f(v)} ∪ {f(vw)|w ∈ V (G),vw ∈ E(G)} for every v ∈ V (G). If f is a k-proper- total-coloring, and for u,v ∈ V (G),uv ∈ E(G), we have Cf(u) = Cf(v), then f is called a k- adjacent-vertex-distinguishing total coloring (k-AV DTC for short). Let χat(G) = min{k|G have a k-adjacent-vertex-distinguishing total coloring}. Then χat(G) is called the adjacent-vertex- distinguishing total chromatic number (AV DTC number for short)...
文摘The total chromatic number xT（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 xT（G） =△（G）＋1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.
文摘A proper total-coloring of graph G is said to be?equitable if the number of elements (vertices and edges) in any?two color classes differ by at most one, which the required?minimum number of colors is called the equitable total chromatic?number. In this paper, we prove some theorems on equitable?total coloring and derive the equitable total chromatic numbers?of Pm V?Sn, Pm V?Fn and Pm V Wn.
文摘A planar graph G is called a i pseudo outerplanar graph if there is a subset V 0V(G),|V 0|=i, such that G-V 0 is an outerplanar graph. In particular, when G-V 0 is a forest, G is called a i pseudo tree. In this paper, the following results are proved: (i) The conjecture on the total coloring is true for all 1 pseudo outerplanar graphs; (ii) χ t(G)=Δ(G)+1 for any 1 pseudo outerplanar graph G with Δ(G)6 and for any 1 pseudo tree G with Δ(G)3, where χ t(G) is the total chromatic number of a graph G .
文摘An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees.The purpose of this paper is to derive exact values of acyclic chromatic number of some graphs.
基金Supported by NNSF of China(61163037,61163054,61363060)
文摘Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） and S（v） S（u） for any two adjacent vertices u and v, then f is called a Smarandachely adjacent vertex distinguishing proper edge col- oring using k colors, or k-SA-edge coloring. The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number, or SA- edge chromatic number for short, and denoted by Xsa（G）. In this paper, we have discussed the SA-edge chromatic number of K4 V Kn.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471131)
文摘Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ? 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.
基金the Natural Science Foundation of Gansu Province （No. 3ZS051-A25-025） the Foundation of Gansu Provincial Department of Education （No. 0501-03）.
文摘Let G = （V, E） be a simple connected graph, and ｜V（G）｜ ≥ 2. Let f be a mapping from V（G） ∪ E（G） to {1,2…, k}. If arbitary uv ∈ E（G）,f（u） ≠ f（v）,f（u） ≠ f（uv）,f（v） ≠ f（uv）; arbitary uv, uw ∈ E（G）（v ≠ w）, f（uv） ≠ f（uw）;arbitary uv ∈ E（G） and u ≠ v, C（u） ≠ C（v）, whereC（u）={f（u）}∪{f（uv）｜uv∈E（G）}.Then f is called a k-adjacent-vertex-distinguishing-proper-total coloring of the graph G（k-AVDTC of G for short）. The number min{k｜k-AVDTC of G} is called the adjacent vertex-distinguishing total chromatic number and denoted by χat（G）. In this paper we prove that if △（G） is at least a particular constant and δ ≥32√△ln△, then χat（G） ≤ △（G） ＋ 10^26 ＋ 2√△ln△.
基金the Xianyang Normal University Foundation for Basic Research(No.06XSYK266)Com~2 MaCKOSEP(R11-1999-054)
文摘A vertex distinguishing equitable total coloring of graph G is a proper total coloring of graph G such that any two distinct vertices＇ coloring sets are not identical and the difference of the elements colored by any two colors is not more than 1. In this paper we shall give vertex distinguishing equitable total chromatic number of join graphs Pn VPn, Cn VCn and prove that they satisfy conjecture 3, namely, the chromatic numbers of vertex distinguishing total and vertex distinguishing equitable total are the same for join graphs Pn V Pn and Cn ∨ Cn.
文摘Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct vertices u and v of V（G）, then f is called a vertex- distinguishing total k-coloring of G. The minimum number k for which there exists a vertex- distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by Xvt（G）. The vertex-disjoint union of two cycles of length n is denoted by 2Cn. We will obtain Xvt（2Cn） in this paper.
基金The Fundamental Research Funds for the Central Universities of China(No.3207013904)
文摘Let G be a simple graph and f be a proper total kcoloring of G. The color set of each vertex v of G is the set of colors appearing on v and the edges incident to v. The coloring f is said to be an adjacent vertex-distinguishing total coloring if the color sets of any two adjacent vertices are distinct. The minimum k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number of G. The join graph of two vertex-disjoint graphs is the graph union of these two graphs together with all the edges that connect the vertices of one graph with the vertices of the other. The adjacent vertex-distinguishing total chromatic numbers of the join graphs of an empty graph of order s and a complete graph of order t are determined.
文摘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 Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.
基金supported by the Natural Science Foundation of Chongqing Science and Technology Commission(Grant No. 2007BB2123)
文摘The total chromatic number χT (G) of a graph G is the minimum number of colors needed to color the edges and the vertices of G so that incident or adjacent elements have distinct colors. We show that if G is a regular graph and d(G) ? 2 3 |V (G)|+ 23 6 , where d(G) denotes the degree of a vertex in G, then χT (G) ? d(G) + 2.
文摘The total chromatic number XT(G) of graph G is the least number of colorsassigned to VE(G) such that no adjacent or incident elements receive the same color.Gived graphs G1,G2, the join of G1 and G2, denoted by G1∨G2, is a graph G, V(G) =V(GI)∪V(G2) and E(G) = E(G1)∪E(G2) ∪{uv | u∈(G1), v ∈ V(G2)}. In this paper, it's proved that if v(G) = v(H), both Gc and Hc contain perfect matching and one of the followings holds: (i)Δ(G) =Δ(H) and there exist edge e∈ E(G), e' E E(H)such that both G-e and H-e' are of Class l; (ii)Δ(G)<Δ(H) and there exixst an edge e ∈E(H) such that H-e is of Class 1, then, the total coloring conjecture is true for graph G ∨H.
基金Supported by the NNSF of China(61163037,61163054)Supported by the Scientific Research Foundation of Ningxia University((E):ndzr09-15)
文摘Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.
