Let fk (∆) be the maximum number of vertices in a planar graph with diameter k and maximum degree ∆. Let gk (∆) be the maximum number of vertices in a maximal planar graph with diameter k and maximum degree ∆. Seyffarth has shown that g2(∆)=「3∆/2+1」 for ∆≥8. Hell and Seyffarth have shown that g2(∆)=「3∆/2+1」 for ∆≥8. We compute the exact maximum number of vertices in a planar graph and a maximal planar graph with diameter two and maximum degree ∆, for ∆＜8.

Kotzig (see Bondy and Murty, Graph Theory with Applications, North-Holland, Amsterdam, 1976) conjectured that there exists no graph with the property that every pair of vertices is connected by a unique path of length k, k>2. Kotzig (Graph Theory and Related Topics, Academic Press, New York, 1979, pp. 358-367) has proved this conjecture for 2<k<9. Xing and Hu (Discrete Math. 135 (1994) 387-393) have proved it for k>11. Here we prove this conjecture for the remaining cases k=9, 10, 11.

We construct a set of three graphs with the property that a 3-connected graph with minimum degree at least 7 is a line graph if and only if it does not contain any of the three graphs as an induced subgraph. We also show that the number of excluded subgraphs cannot be reduced no matter what connectivity is specified.

Calculating the crossing number of a given graph is, in general, an elusive problem. As Garey and Johnson have proved, the problem of determining the crossing number of an arbitrary graph is NP-complete (Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (1983) 312-316). The crossing numbers of very few families of graphs are known exactly. Richter and Salazar (The crossing number of P(N, 3), Graphs and Combinatorics 18 (2) (2002) 381-394) have studied the crossing number of the generalized Petersen graph P(n, 3) and proved that cr(P (3h, 3))=h (h≥4); cr(P (3h+1, 3))=h+3 (h≥3); cr(P (3h+2, 3))=h+2 (h≥3). In this paper, we study the crossing number of the circulant graph C(n; {1, 3}) and prove that cr(C(n; {1, 3}))=「n/3」+n mod 3 (n≥8).

Given a set of graphs Ψ={G1, G2, …, Gk}, let ex(n; Ψ) denote the greatest size of a graph with order n that contains no subgraph isomorphic to some Gi, 1≤i≤k. One of the main classes of problems in extremal graph theory, known as Tuŕan-type problems, is for given n, Ψ to determine explicitly the function ex (n, Ψ), or to find its asymptotic behavior. Yang Yuansheng investigated the values of ex(n, Ψ) for Ψ={C4} (UTILITAS MATHEMATICA, 41 (1992), 204-210), Garnick investigated them for Ψ={C3, C4} (Journal of Graph Theory, vol.17, no.5 (1993), 633-645) and Alabdullatif investigated them for Ψ={Cn-k+1, …, Cn and Ψ={Pn-k+1,…, Pn}, (1≤k≤n-2) (Bull. Inst. Combin. Appl., 25 (1999)41-52). This paper investigates the values of ex (n, Ψ) forΨ={C3, C4, C5}, n≤42.

Given t(≥2) cycles Cn of length n≥3, each with a fixed vertex vi0, i=1, 2,…, t, let C(t)n denote the graph obtained from the union of the t cycles by identifying the t fixed vertices(v10=v20=…=vt0). Koh et al. conjectured that C(t)n is graceful if and only if nt≡0,3 (mod 4). The conjecture has been shown true for n=3, 6, 4k. In this paper, the conjecture is shown to be true for n=5.

In this article, we show that the disjoint union C2k UC2j+1 of cycles C2k and C2j+1 (k

The path layer matrix of a graph G contains quantitative information about all possible paths in G. The entry (i; j ) of this matrix is the number of paths in G having initial vertex i and length j. It is known that there are 4-regular graphs on 44 vertices having the same path layer matrix[Y. Yuansheng, L. Jianhua, and W. Chunli, J Graph Theory 39(2002) 219-221] graphs with cut-vertices on 14 vertices having the same path layer matrix [A. A. Dobrynin, Vyčisl. sistemy, Novosibirsk 119(1987) 13-33] and graphs without cut-vertices on 31 vertices having the same path layer matrix [A. A. Dobrynin, J Graph Theory 38(2001) 177-182]. In this article, a pair of 4-regular graphs without cut-vertices on 18 vertices having the same path layer matrix are constructed, improving the upper bound for the least order of 4-regular graphs having the same path layer matrix from 44 to 18 and the upper bound for the least order of graphs without cut-vertices having the same path layer matrix from 31 to 18.

The path layer matrix of graph G contains quantitative information about all paths in G. The entry (i,j) in this matrix is the number of simple paths in G having initial vertexi and length j. Some new upper bounds for r-regular graphs with the same path layer matrix are presented for r=4, 5, 6.

利用计算机对图的交叉数进行研究，给出利用分支界限法计算图的交叉数的算法CCN（calculate crossing Number），并利用该算法计算出n≤12的所有四正则图的交叉数，以及n≤16的随机四正则的交叉数。同时计算出n≤12的所有四正则图的平均交叉数Aac（n），和n≤16的随机四正则图的平均交叉数Arc（n），根据计算结果提出四正则图的平均交叉数为O（n2）的猜想。