Another step towards proving a conjecture by Plummer and Toft
نویسندگان
چکیده
A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(G) ≤ ∆ ∗+2 for any 3-connected plane graph G with maximum face degree ∆∗. It is known that the conjecture holds true for ∆∗ ≤ 4 and ∆∗ ≥ 24. The validity of the conjecture is proved in the paper for ∆∗ ≥ 18.
منابع مشابه
On a special case of Hadwiger's conjecture
Hadwiger’s Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α(G) = 2. We present some results in this special case.
متن کاملIndependence number and clique minors
Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem o...
متن کاملConnected matchings in chordal bipartite graphs
A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft [13] introduced connected matchings and proved that, given a positive integer k, determining whether a graph has a connected matching of size at least k is NPcomplete. Cameron [4] proved that ...
متن کاملA note on cyclic chromatic number
A cyclic colouring of a graph G embedded in a surface is a vertex colouring of G in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(G) ≤ ∆∗ + 2 for any 3-connected plane graph G with maximum face degree ∆∗. It is known that th...
متن کاملImprovements of the theorem of Duchet and Meyniel on Hadwiger's conjecture
Probably the best-known among the remaining unsolved problems in graph theory is Hadwiger’s Conjecture: If χ(G) denotes the chromatic number of graph G, then G has the complete graph Kχ(G) as a minor. The following would immediately follow from the truth of Hadwiger’s Conjecture: Conjecture: If G has n vertices and if α(G) denotes the independence number of G, then G has Kdn/α(G)e as a minor. I...
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:
- Discrete Mathematics
دوره 310 شماره
صفحات -
تاریخ انتشار 2010