Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that an...Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.展开更多
The minimum number of total independent partition sets of V ∪ E of graph G（V,E） is called the total chromatic number of G denoted by χt（G）. If the difference of the numbers of any two total independent partition...The minimum number of total independent partition sets of V ∪ E of graph G（V,E） is called the total chromatic number of G denoted by χt（G）. If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one＇, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χet（G）. In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.展开更多
Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P...Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P^((3))_(1)∨K_((2))^(2n))were presented.For natural number n∈N,n≥1,the two graphs are all graceful graphs,where P^((1))_(2n),P^((2))_(2n)are even-vertices path,P_(2n+1)is odd-vertices path,K_(2n),K^((1))_(2n),K^((2))_(2n)are the complement of graph K_(2 n),G_(1)∨G_(2)is the join graph of G_(1)and G_(2).展开更多
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 ...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 △≥ 9, then ）χ″（G） =△＋ 1. In this paper, we prove that if O is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ″（G） =- 9.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No. 10001005) and by RFDP of China.
文摘Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.
基金Supported by the National Natural Science Foundation of China(No.10771091)
文摘The minimum number of total independent partition sets of V ∪ E of graph G（V,E） is called the total chromatic number of G denoted by χt（G）. If the difference of the numbers of any two total independent partition sets of V ∪ E is no more than one＇, then the minimum number of total independent partition sets of V ∪ E is called the equitable total chromatic number of G, denoted by χet（G）. In this paper, we obtain the equitable total chromatic number of the join graph of fan and wheel with the same order.
基金the National Natural Science Foundation of China(11702094)the Fundamental Research Funds for the Central University(3142015045)。
文摘Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P^((3))_(1)∨K_((2))^(2n))were presented.For natural number n∈N,n≥1,the two graphs are all graceful graphs,where P^((1))_(2n),P^((2))_(2n)are even-vertices path,P_(2n+1)is odd-vertices path,K_(2n),K^((1))_(2n),K^((2))_(2n)are the complement of graph K_(2 n),G_(1)∨G_(2)is the join graph of G_(1)and G_(2).
基金Supported by Natural Science Foundation of Shandong Province(Grant No.ZR2013AM001)
文摘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 △≥ 9, then ）χ″（G） =△＋ 1. In this paper, we prove that if O is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ″（G） =- 9.
