A Strengthening of Brooks' Theorem for Line Graphs
نویسنده
چکیده
We prove that if G is the line graph of a multigraph, then the chromatic number χ(G) of G is at most max {
منابع مشابه
Every 4-Colorable Graph With Maximum Degree 4 Has an Equitable 4-Coloring
Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. If true, this would be a joint strengthening of the Hajnal-Szemerédi Theorem and Brooks' Theorem. Chen, Lih, and Wu proved that their conjecture holds for r = 3. In this paper we study properties of the hypothetical minimum count...
متن کاملEquitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture
Chen, Lih, and Wu conjectured that for r≥3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. If true, this would be a strengthening of the Hajnal–Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r≥6 the conjecture says the following: If an r-colorable graph G with maximum degree...
متن کاملA Local Strengthening of Reed's Omega, Delta, Chi Conjecture for Quasi-line Graphs
Reed’s ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ d 12 (∆ + 1 + ω)e; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algori...
متن کاملColoring signed graphs using DFS
We show that depth first search can be used to give a proper coloring of connected signed graphs G using at most ∆(G) colors, provided G is different from a balanced complete graph, a balanced cycle of odd length, and an unbalanced cycle of even length, thus giving a new, short proof to the generalization of Brooks’ theorem to signed graphs, first proved by Máčajová, Raspaud, and Škoviera.
متن کاملA local strengthening of Reed’s ω, ∆, and χ conjecture for quasi-line graphs
Reed’s ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ d 12 (∆ + 1 + ω)e; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algori...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 18 شماره
صفحات -
تاریخ انتشار 2011