منابع مشابه
Hajós' conjecture for line graphs
We prove that, if a graph G (without multiple edges) has maximum degree d and edge-chromatic number d + 1, then G contains two distinct vertices x, y and a collection of d pairwise edge-disjoint paths between x and y. In particular, the line graph of G satisfies Hajós’ conjecture. © 2006 Elsevier Inc. All rights reserved.
متن کاملTriangulations and the Hajós Conjecture
The Hajós Conjecture was disproved in 1979 by Catlin. Recently, Thomassen showed that there are many ways that Hajós conjecture can go wrong. On the other hand, he observed that locally planar graphs and triangulations of the projective plane and the torus satisfy Hajós Conjecture, and he conjectured that the same holds for arbitrary triangulations of closed surfaces. In this note we disprove t...
متن کاملOn the conjecture of Hajós
Let G=G(n) be a graph of n vertices. Let X=X(G) denote its chromatic number and a=o(G) the largest integer I so that G contains a subdivision of K, i.e . a(G)=1 is the largest integer such that G contains a subgraph homeomorphic with complete graph of l vertices . Let us put H(G)= X(G) and H(n)=max H(G(n)) o (G) G(n) Hajós [10] conjectured that H(n)=1 and Catlin, [2] recently disproved the conj...
متن کاملOn a Coloring Conjecture of Hajós
Hajós conjectured that graphs containing no subdivision of K5 are 4-colorable. It is shown in [15] that if there is a counterexample to this conjecture then any minimum such counterexample must be 4-connected. In this paper, we further show that if G is a minimum counterexample to Hajós’ conjecture and S is a 4-cut in G then G− S has exactly two components. AMS Subject Classification: 05C15, 05...
متن کاملA cycle decomposition conjecture for Eulerian graphs
A classic theorem of Veblen states that a connected graph G has a cycle decomposition if and only if G is Eulerian. The number of odd cycles in a cycle decomposition of an Eulerian graph G is therefore even if and only if G has even size. It is conjectured that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph G with m edges is a and the maximum number of odd cyc...
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ژورنال
عنوان ژورنال: European Journal of Combinatorics
سال: 2010
ISSN: 0195-6698
DOI: 10.1016/j.ejc.2009.08.008