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1.
Nešetřil and Sopena introduced the concept of oriented game chromatic number. They asked whether the oriented game chromatic
number of partial k-trees was bounded. Here we answer their question positively.
Received: January 12, 2001 Final version received: February 25, 2002 相似文献
2.
A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here. 相似文献
3.
It is shown that if in a simple graph G of order n the sum of degrees of any three pairwise non-adjacent vertices is at least n, then there are two cycles (or one cycle and an edge or a vertex) of GF that contain all the vertices. © 1995 John Wiley & Sons, Inc. 相似文献
4.
Zsolt Tuza 《Graphs and Combinatorics》1990,6(1):51-59
Coloring the vertex set of a graphG with positive integers, thechromatic sum (G) ofG is the minimum sum of colors in a proper coloring. Thestrength ofG is the largest integer that occurs in every coloring whose total is(G). Proving a conjecture of Kubicka and Schwenk, we show that every tree of strengths has at least ((2 +
)
s–1 – (2 –
)
s–1)/
vertices (s 2). Surprisingly, this extremal result follows from a topological property of trees. Namely, for everys 3 there exist precisely two treesT
s
andR
s
such that every tree of strength at leasts is edge-contractible toT
s
orR
s
. 相似文献
5.
Letm 3 andk 1 be two given integers. Asub-k-coloring of [n] = {1, 2,...,n} is an assignment of colors to the numbers of [n] in which each color is used at mostk times. Call an
arainbow set if no two of its elements have the same color. Thesub-k-Ramsey number sr(m, k) is defined as the minimumn such that every sub-k-coloring of [n] contains a rainbow arithmetic progression ofm terms. We prove that((k – 1)m
2/logmk) sr(m, k) O((k – 1)m
2 logmk) asm , and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.Research supported in part by Allon Fellowship and by a Bat Sheva de-Rothschild grant.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant No. 1-3-86-264. 相似文献
6.
The intersection dimension of a graphG with respect to a classA of graphs is the minimumk such thatG is the intersection of at mostk graphs on vertex setV(G) each of which belongs toA. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) ifA is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect toA is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.:On grid intersection graphs, Discrete Math.87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs. 相似文献
7.
We discuss problems and results on the maximum number of colors in combinatorial structures under the assumption that no totally multicolored sets of a specified type occur. 相似文献
8.
9.
A vertex v of a graph G is called groupie if the average degree tv of all neighbors of v in G is not smaller than the average degree tG of G. Every graph contains a groupie vertex; the problem of whether or not every simple graph on ≧2 vertices has at least two groupie vertices turned out to be surprisingly difficult. We present various sufficient conditions for a simple graph to contain at least two groupie vertices. Further, we investigate the function f(n) = max minv (tv/tG), where the maximum ranges over all simple graphs on n vertices, and prove that f(n) = 1/42n + o(1). The corresponding result for multigraphs is in sharp contrast with the above. We also characterize trees in which the local average degree tv is constant. 相似文献
10.
Minimal colorings for properly colored subgraphs 总被引:1,自引:0,他引:1
We give conditions on the minimum numberk of colors, sufficient for the existence of given types of properly edge-colored subgraphs in ak-edge-colored complete graph. The types of subgraphs we study include families of internally pairwise vertex-disjoint paths with common endpoints, hamiltonian paths and hamiltonian cycles, cycles with a given lower bound of their length, spanning trees, stars, and cliques. Throughout the paper, related conjectures are proposed.Research supported by a European Union PECO grant under identification number CIPA3510PL929589, while on leave at University Paris-XI 相似文献