An Ore-type theorem on equitable coloring
نویسندگان
چکیده
A proper vertex coloring of a graph is equitable if the sizes of its color classes differ by at most one. In this paper, we prove that if G is a graph such that for each edge xy ∈ E(G), the sum d(x)+ d(y) of the degrees of its ends is at most 2r + 1, then G has an equitable coloring with r + 1 colors. This extends the Hajnal–Szemerédi Theorem on graphs with maximum degree r and a recent conjecture by Kostochka and Yu. We also pose an Ore-type version of the Chen–Lih–Wu Conjecture and prove a very partial case of it. © 2007 Elsevier Inc. All rights reserved.
منابع مشابه
Sharpening an Ore-type Version of the Corrádi-hajnal Theorem
In 1963, Corrádi and Hajnal proved that for all k ≥ 1 and n ≥ 3k, every (simple) graph G on n vertices with minimum degree δ(G) ≥ 2k contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corrádi-Hajnal Theorem: For all k ≥ 1 and n ≥ 3k, every graph G on n vertices contains k disjoint cycles, provided that d(x) + d(y) ≥ 4k − 1 for...
متن کاملExtremal graph packing problems: Ore-type versus Dirac-type
We discuss recent progress and unsolved problems concerning extremal graph packing, emphasizing connections between Dirac-type and Ore-type problems. Extra attention is paid to coloring, and especially equitable coloring, of graphs.
متن کاملEquitable defective coloring of sparse planar graphs
A graph has an equitable, defective k-coloring (an ED-k-coloring) if there is a kcoloring of V (G) that is defective (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). A graph may have an ED-kcoloring, but no ED-(k + 1)-coloring. In this paper, we prove that planar graphs with minimum degree at least 2 and girth a...
متن کاملEquitable Coloring of k-Uniform Hypergraphs
Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts, such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is dn/re or bn/rc. We prove that for all a ≥ 1, if the maximum degree of H satisfies ∆(H) ≤ k then H has an equitable coloring with k a ln k (1 − ok(1)) parts. In particul...
متن کاملEquitable Colourings Of D-Degenerate Graphs
In many applications of graph colouring the sizes of colour classes should not be too large. For example, in scheduling jobs (some of which could be performed at the same time), it is not good if the resulting schedule requires many jobs to occur at some specific time. An application of this type is discussed in [8]. A possible formalization of this restriction is the notion of equitable colour...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Comb. Theory, Ser. B
دوره 98 شماره
صفحات -
تاریخ انتشار 2008