The circular chromatic number of a graph is an important parameter of a graph. The distance graph G(Z,D) , with a distance set D , is the infinite graph with vertex set Z={0,±1,±2,...} in which tw...The circular chromatic number of a graph is an important parameter of a graph. The distance graph G(Z,D) , with a distance set D , is the infinite graph with vertex set Z={0,±1,±2,...} in which two vertices x and y are adjacent iff y-x∈D . This paper determines the circular chromatic numbers of two classes of distance graphs G(Z,D m,k,k+1 ) and G(Z,D m,k,k+1,k+2 ).展开更多
For a general graph G, M（G） denotes its Mycielski graph. This article gives a number of new sufficient conditions for G to have the circular chromatic number xc（M（G）） equals to the chromatic number x（M（G））, ...For a general graph G, M（G） denotes its Mycielski graph. This article gives a number of new sufficient conditions for G to have the circular chromatic number xc（M（G）） equals to the chromatic number x（M（G））, which have improved some best sufficient conditions published up to date.展开更多
The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, t...The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with △（G）≥｜G｜ - 2≥9 has Xef（G） = △（G）.展开更多
A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality...A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.展开更多
The circular chromatic number of a graph is a natural generalization of the chromatic number. Circular chromatic number contains more information about the structure of a graph than chromatic number does. In this pape...The circular chromatic number of a graph is a natural generalization of the chromatic number. Circular chromatic number contains more information about the structure of a graph than chromatic number does. In this paper we obtain the circular chromatic numbers of special graphs such as C t k and C t k-v, and give a simple proof of the circular chromatic number of H m,n .展开更多
For two integers k and d with （k, d） = 1 and k≥2d, let G^dk be the graph with vertex set {0,1,…k - 1 } in which ij is an edge if and only if d≤｜ i -j I｜≤k - d. The circular chromatic number χc（G） of a graph...For two integers k and d with （k, d） = 1 and k≥2d, let G^dk be the graph with vertex set {0,1,…k - 1 } in which ij is an edge if and only if d≤｜ i -j I｜≤k - d. The circular chromatic number χc（G） of a graph G is the minimum of k/d for which G admits a homomorphism to G^dk. The relationship between χc（ G- v） and χc （G）is investigated. In particular, the circular chromatic number of G^dk - v for any vertex v is determined. Some graphs withx χc（G - v） =χc（G） - 1 for any vertex v and with certain properties are presented. Some lower bounds for the circular chromatic number of a graph are studied, and a necessary and sufficient condition under which the circular chromatic number of a graph attains the lower bound χ- 1 ＋ 1/α is proved, where χ is the chromatic number of G and a is its independence number.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
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...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.展开更多
Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper...Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper, it is proved that △ ＋ 1 ≤ x（G^2） ≤△ ＋ 2, and x（G^2） = A ＋ 2 if and only if G is Q, where A represents the maximum degree of G.展开更多
Let G be a maximal outerplane graph and X0(G) the complete chromatic number of G. This paper determines exactly X0(G) for △(G)≠5 and proves 6≤X0.(G)≤7 for △(G) = 5, where △(G) is the maximum degree of vertices o...Let G be a maximal outerplane graph and X0(G) the complete chromatic number of G. This paper determines exactly X0(G) for △(G)≠5 and proves 6≤X0.(G)≤7 for △(G) = 5, where △(G) is the maximum degree of vertices of G.展开更多
Wang Wei-fan[1] proved that the edge-face chromatic number of a 2-connected 1-tree with the maximum degree is not less than 6 is its maximum degree, and he conjectured that it is true when the maximum degree is 5. Thi...Wang Wei-fan[1] proved that the edge-face chromatic number of a 2-connected 1-tree with the maximum degree is not less than 6 is its maximum degree, and he conjectured that it is true when the maximum degree is 5. This paper proves the conjecture.展开更多
For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been wid...For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been widely studied. Moreover, the form of minimum such an f is also concerned. A result of Schiermeyer shows that every -free graph G with clique number has . Chudnovsky and Sivaraman proved that every -free with clique number graph is -colorable. In this paper, for any -free graph G with clique number , we prove that . The main methods in the proof are set partition and induction.展开更多
Let G be a graph.We useχ(G)andω(G)to denote the chromatic number and clique number of G respectively.A P_(5)is a path on 5 vertices,and an HVN is a K_(4)together with one more vertex which is adjacent to exactly two...Let G be a graph.We useχ(G)andω(G)to denote the chromatic number and clique number of G respectively.A P_(5)is a path on 5 vertices,and an HVN is a K_(4)together with one more vertex which is adjacent to exactly two vertices of K_(4).Combining with some known result,in this paper we show that if G is(P_(5),HVN)-free,thenχ(G)≤max{min{16,ω(G)+3},ω(G)+1}.This upper bound is almost sharp.展开更多
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
In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the doub...In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the double inductions and the method of exchanging colors from the aspect of configuration property. For series-parallel graphs of △(G) ≥ 5, △(G) ≤ x'as(G) ≤ △(G) + 1. Moreover, x'as(G) = △(G) + 1 if and only if it has two adjacent vertices of maximum degree, where △(G) and X'as(G) denote the maximum degree and the adjacent strong edge chromatic number of graph G respectively.展开更多
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 ...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 v, where uv ∈E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ＇αα（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＇αα（G）≤32△.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
文摘The circular chromatic number of a graph is an important parameter of a graph. The distance graph G(Z,D) , with a distance set D , is the infinite graph with vertex set Z={0,±1,±2,...} in which two vertices x and y are adjacent iff y-x∈D . This paper determines the circular chromatic numbers of two classes of distance graphs G(Z,D m,k,k+1 ) and G(Z,D m,k,k+1,k+2 ).
基金Supported by National Science Foundation of China (10371048)the Science Foundation of Three Gorges University.
文摘For a general graph G, M（G） denotes its Mycielski graph. This article gives a number of new sufficient conditions for G to have the circular chromatic number xc（M（G）） equals to the chromatic number x（M（G））, which have improved some best sufficient conditions published up to date.
基金This research is supported by NNSF of China(40301037, 10471131)
文摘The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with △（G）≥｜G｜ - 2≥9 has Xef（G） = △（G）.
文摘A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.
文摘The circular chromatic number of a graph is a natural generalization of the chromatic number. Circular chromatic number contains more information about the structure of a graph than chromatic number does. In this paper we obtain the circular chromatic numbers of special graphs such as C t k and C t k-v, and give a simple proof of the circular chromatic number of H m,n .
基金The National Natural Science Foundation of China(No.10671033)
文摘For two integers k and d with （k, d） = 1 and k≥2d, let G^dk be the graph with vertex set {0,1,…k - 1 } in which ij is an edge if and only if d≤｜ i -j I｜≤k - d. The circular chromatic number χc（G） of a graph G is the minimum of k/d for which G admits a homomorphism to G^dk. The relationship between χc（ G- v） and χc （G）is investigated. In particular, the circular chromatic number of G^dk - v for any vertex v is determined. Some graphs withx χc（G - v） =χc（G） - 1 for any vertex v and with certain properties are presented. Some lower bounds for the circular chromatic number of a graph are studied, and a necessary and sufficient condition under which the circular chromatic number of a graph attains the lower bound χ- 1 ＋ 1/α is proved, where χ is the chromatic number of G and a is its independence number.
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.
文摘Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper, it is proved that △ ＋ 1 ≤ x（G^2） ≤△ ＋ 2, and x（G^2） = A ＋ 2 if and only if G is Q, where A represents the maximum degree of G.
基金Project supported by the Vatural SCience Foundation of LNEC.
文摘Let G be a maximal outerplane graph and X0(G) the complete chromatic number of G. This paper determines exactly X0(G) for △(G)≠5 and proves 6≤X0.(G)≤7 for △(G) = 5, where △(G) is the maximum degree of vertices of G.
文摘Wang Wei-fan[1] proved that the edge-face chromatic number of a 2-connected 1-tree with the maximum degree is not less than 6 is its maximum degree, and he conjectured that it is true when the maximum degree is 5. This paper proves the conjecture.
文摘For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been widely studied. Moreover, the form of minimum such an f is also concerned. A result of Schiermeyer shows that every -free graph G with clique number has . Chudnovsky and Sivaraman proved that every -free with clique number graph is -colorable. In this paper, for any -free graph G with clique number , we prove that . The main methods in the proof are set partition and induction.
基金supported by the National Natural Science Foundation of China(No.12101117)Natural Science Foundation of Jiangsu Province(No.BK20200344)。
文摘Let G be a graph.We useχ(G)andω(G)to denote the chromatic number and clique number of G respectively.A P_(5)is a path on 5 vertices,and an HVN is a K_(4)together with one more vertex which is adjacent to exactly two vertices of K_(4).Combining with some known result,in this paper we show that if G is(P_(5),HVN)-free,thenχ(G)≤max{min{16,ω(G)+3},ω(G)+1}.This upper bound is almost sharp.
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National Natural Science Foundation of China (60103021, 60274026)
文摘In this paper, we will study the adjacent strong edge coloring of series-parallel graphs, and prove that series-parallel graphs of △(G) = 3 and 4 satisfy the conjecture of adjacent strong edge coloring using the double inductions and the method of exchanging colors from the aspect of configuration property. For series-parallel graphs of △(G) ≥ 5, △(G) ≤ x'as(G) ≤ △(G) + 1. Moreover, x'as(G) = △(G) + 1 if and only if it has two adjacent vertices of maximum degree, where △(G) and X'as(G) denote the maximum degree and the adjacent strong edge chromatic number of graph G respectively.
基金Supported by the Natural Science Foundation of Gansu Province(3ZS051-A25-025)
文摘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 v, where uv ∈E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ＇αα（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＇αα（G）≤32△.
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.
