It is shown that K 4(i,j,l,k,m,n) is chromatically unique if three numbers among i,j,l,k,m,n have the same value and the other three numbers are not equal but larger than that value.
Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r ...Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.展开更多
By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph（the Theorem 1）, a first class hi-cycle graph and a second class bicycle graph...By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph（the Theorem 1）, a first class hi-cycle graph and a second class bicycle graph（the Theorem 2）, respectively.展开更多
Two graphs are defined to be adjointly equivalent if and only if their complements are chromatically equivalent. Using the properties of the adjoint polynomials and the fourth character R4（G）, the adjoint equivalenc...Two graphs are defined to be adjointly equivalent if and only if their complements are chromatically equivalent. Using the properties of the adjoint polynomials and the fourth character R4（G）, the adjoint equivalence class of graph Bn-8,l,4 is determined. According to the relations between adjoint polynomial and chromatic polynomial, we also simultaneously determine the chromatic equivalence class of Bn-8,l,4 that is the complement of Bn-8,l,4.展开更多
In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u...In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u i≠4k+2.展开更多
In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u...In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u i≠4k+2.展开更多
The parameter R（G） is the function about the front three coeffcients of the adjoint polynomial of graph G. In the paper, the range of R（G） is given when β（G） 〈 β（Dn）, where β（G） is the minimum root of th...The parameter R（G） is the function about the front three coeffcients of the adjoint polynomial of graph G. In the paper, the range of R（G） is given when β（G） 〈 β（Dn）, where β（G） is the minimum root of the adjoint polynomial of graph G and the chromatically equivalent classification of tDn is completely depicted.Furthermore, a sufficient and necessary condition for the class of graphs to be chromatically unique is obtained.展开更多
文摘It is shown that K 4(i,j,l,k,m,n) is chromatically unique if three numbers among i,j,l,k,m,n have the same value and the other three numbers are not equal but larger than that value.
文摘Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.
基金Supported by the NNSF of China(10861009)Supported by the Ministry of Education Science and Technology Item of China(206156)
文摘By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph（the Theorem 1）, a first class hi-cycle graph and a second class bicycle graph（the Theorem 2）, respectively.
基金Supported by the National Natural Science Foundation of China(Grant No.11161037)the Science Found of Qinghai Province(Grant No.2011-z-907)
文摘Two graphs are defined to be adjointly equivalent if and only if their complements are chromatically equivalent. Using the properties of the adjoint polynomials and the fourth character R4（G）, the adjoint equivalence class of graph Bn-8,l,4 is determined. According to the relations between adjoint polynomial and chromatic polynomial, we also simultaneously determine the chromatic equivalence class of Bn-8,l,4 that is the complement of Bn-8,l,4.
文摘In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u i≠4k+2.
文摘In this paper, using the properties of chromatic polynomial, we discuss the color-partition of the complement of lK 1∪(∪C u i),and characterize the graph with the same color-partition as the class graph under u i≠4k+2.
基金Supported by the National Science Foundation of China(10761008)Supported by the Science Foundation of the State Education Ministry of China(205170)
文摘The parameter R（G） is the function about the front three coeffcients of the adjoint polynomial of graph G. In the paper, the range of R（G） is given when β（G） 〈 β（Dn）, where β（G） is the minimum root of the adjoint polynomial of graph G and the chromatically equivalent classification of tDn is completely depicted.Furthermore, a sufficient and necessary condition for the class of graphs to be chromatically unique is obtained.
