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1.
Subgraph distances in graphs defined by edge transfers 总被引:1,自引:0,他引:1
Gary Chartrand H ctor Hevia Elzbieta B. Jarrett Michelle Schultz 《Discrete Mathematics》1997,170(1-3):63-79
For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{Jk(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence. 相似文献
2.
Stefan A. Burr 《Discrete Mathematics》1988,70(3):277-293
Let
be a fixed finite set of connected graphs. Results are given which, in principle, permit the Ramsey number r(G, H) to be evaluated exactly when G and H are sufficiently large disjoint unions of graphs taken from
. Such evaluations are often possible in practice, as shown by several examples. For instance, when m and n are large, and mn, 相似文献
r(mKk, nKl)=(k − 1)m+ln+r(Kk−1, Kl−1)−2.
3.
Let G be a k-regular vertex transitive graph with connectivity κ(G)=k and let mk(G) be the number of vertex cuts with k vertices. Define m(n,k)=min{mk(G): GTn,k}, where Tn,k denotes the set of all k-regular vertex transitive graphs on n vertices with κ(G)=k. In this paper, we determine the exact values of m(n,k). 相似文献
4.
Hong-Jian Lai 《Discrete Mathematics》2001,230(1-3):63-69
For an integer l0, define
to be the family of graphs such that
if and only if for any edge subset XE(G) with |X|l, G has a spanning eulerian subgraph H with XE(H). The graphs in
are known as supereulerian graphs. Let f(l) be the minimum value of k such that every k-edge-connected graph is in
. Jaeger and Catlin independently proved f(0)=4. We shall determine f(l) for all values of l0. Another problem concerning the existence of eulerian subgraphs containing given edges is also discussed, and former results in [J. Graph Theory 1 (1977) 79–84] and [J. Graph Theory 3 (1979) 91–93] are extended. 相似文献
5.
Some results on integral sum graphs 总被引:1,自引:0,他引:1
Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(Kn−E(Kr)) for r2n/3−1, (ii) obtain a lower bound for ζ(Kn−E(Kr)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(Kr,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of Kr,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211). 相似文献
6.
R. J. Faudree R. J. Gould M. S. Jacobson J. Lehel L. M. Lesniak 《Discrete Mathematics》1996,150(1-3):103-113
The k-spectrum sk(G) of a graph G is the set of all positive integers that occur as the size of an induced k-vertex subgraph of G. In this paper we determine the minimum order and size of a graph G with sk (G) = {0, 1, …,(2k)} and consider the more general question of describing those sets S {0,1, … ,(2k)} such that S = sk(G) for some graph G. 相似文献
7.
Length-bounded disjoint paths in planar graphs 总被引:1,自引:0,他引:1
The following problem is considered: given: an undirected planar graph G=(V,E) embedded in
, distinct pairs of vertices {r1,s1},…,{rk,sk} of G adjacent to the unbounded face, positive integers b1,…,bk and a function
; find: pairwise vertex-disjoint paths P1,…,Pk such that for each i=1,…,k, Pi is a ri−si-path and the sum of the l-length of all edges in Pi is at most bi. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2. 相似文献
8.
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ'_s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2 k. Yang and Zhu [J. Graph Theory, 83, 334–339(2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4 kΔ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤ 4 kΔ-2 k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most(4 k-1)Δ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤(4 k-1)Δ-2 k + 1. 相似文献
9.
The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V1, V2,…, Va such that for each i at most d vertices in V1 … Vi have neighbors in Vi+1 and r(Kk, Vi) p | V(G) |, where Vi denotes the subgraph of G induced by Vi. Then there exists a number c depending only on p, d, and k such that r(Kk, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4. 相似文献
10.
Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |NG(x1)NG(x2)NG(xk)|an/(a+b) for any independent subset {x1,x2,…,xk} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense. 相似文献
11.
Xuding Zhu 《Discrete Mathematics》1998,190(1-3):215-222
Suppose G is a graph. The chromatic Ramsey number rc(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let Mn = min[rc(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that Mn = (n − 1)2 + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M6 and M7. 相似文献
12.
If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=(a1,a2,…,ak), in which ai denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph Kn,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of Kn,n into multicolored trees. 相似文献
13.
14.
Ludwin A.BASILIO-HERNáNDEZ Walter CARBALLOSA Jesús LEA?OS José M.SIGARRETA 《数学学报(英文版)》2019,35(3):338-354
We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G. 相似文献
15.
C. A. Barefoot L. H. Clark R. C. Entringer T. D. Porter L. A. Sz kely Zs. Tuza 《Discrete Mathematics》1996,150(1-3):31-48
A graph G is called Ck-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in Ck-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + c1n/k and n + c2n/k for some positive constants c1 and C2. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás. 相似文献
16.
Oleg V. Borodin 《Discrete Mathematics》1992,100(1-3):281-289
Let G be a plane graph, and let χk(G) be the minimum number of colors to color the vertices of G so that every two of them which lie in the boundary of the same face of the size at most k, receive different colors. In 1966, Ore and Plummer proved that χk(G)2k for any k3. It is also known that χ3(G)4 (Appel and Haken, 1976) and χ4(G)6 (Borodin, 1984). The result in the present paper is: χ5(G)9, χ6(G)11, χ7(G)12, and χk(G)2k − 3 if k8. 相似文献
17.
《Discrete Mathematics》1999,200(1-3):61-77
We say (n, e) → (m, f), an (m, f) subgraph is forced, if every n-vertex graph of size e has an m-vertex spanned subgraph with f edges. For example, as Turán proved, (n,e)→(k,(k2)) for e> tk − 1(n) and (n,e) (k2)), otherwise. We give a number of constructions showing that forced pairs are rare. Using tools of extremal graph theory we also show infinitely many positive cases. Several problems remain open. 相似文献
18.
The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u1,v1) is adjacent to (u2,v2) if either u1 is adjacent to u2 in G or u1=u2 and v1 is adjacent to v2 in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define wG(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of wG(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph. 相似文献
19.
For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number χ(G) = k can be properly vertex colored by k colors so that for each pair of vertices xo and xp in any color class there exist vertices x1, x2, …, xp-1 of the same class with dist(xi, xi+1) f(k) for each i, 0 i p − 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.
We prove that f(k) < 12k when the order of the graph is k(k − 2) + 1. 相似文献
20.
Stephen Olariu 《Discrete Mathematics》1990,80(3):281-296
An edge uv of a graph G is called a wing if there exists a chordless path with vertices u, v, x, y and edges uv, vx, xy. The wing-graph W(G) of a graph G is a graph having the same vertex set as G; uv is an edge in W(G) if and only if uv is a wing in G. A graph G is saturated if G is isomorphic to W(G). A star-cutset in a graph G is a non-empty set of vertices such that G — C is disconnected and some vertex in C is adjacent to all the remaining vertices in C. V. Chvátal proposed to call a graph unbreakable if neither G nor its complement contain a star-cutset. We establish several properties of unbreakable graphs using the notions of wings and saturation. In particular, we obtain seven equivalent versions of the Strong Perfect Graph Conjecture. 相似文献