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Chromatic Index Critical Graphs of Orders 11 and 12
, 1997
"... A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results thi ..."
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A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three nontrivial chromaticindex critical graphs of order 12. 1 Introduction A famous theorem of Vizing [20] states that the chromatic index Ø 0 (G) of a simple graph G is \Delta(G) or \Delta(G) + 1, where \Delta(G) denotes the maximum vertex degree in G. A graph G is class 1 if Ø 0 (G) = \Delta(G) and it is class 2 otherwise. A class 2 graph G is (chromatic index) critical if Ø 0 (G \Gamma e) ! Ø 0 (G) for each edge e of G. If we want to stress the maximum vertex degree of a critical graph G we say G is \Delta(G)critical. Critical graphs of odd order are easy to construct while not much is known about critical graphs of even order. One reas...
NowhereZero KFlows of Supergraphs
"... Let G be a 2edgeconnected graph with o vertices of odd degree. It is wellknown that one should (and can) add o 2 edges to G in order to obtain a graph which admits a nowherezero 2flow. We prove that one can add to G asetof## o 4 #, # 1 2 # o 5 ##,and # 1 2 # o 7 ## edges such t ..."
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Let G be a 2edgeconnected graph with o vertices of odd degree. It is wellknown that one should (and can) add o 2 edges to G in order to obtain a graph which admits a nowherezero 2flow. We prove that one can add to G asetof## o 4 #, # 1 2 # o 5 ##,and # 1 2 # o 7 ## edges such that the resulting graph admits a nowherezero 3flow, 4flow, and 5flow, respectively. 1
ChromaticIndex Critical Graphs of Even Order
, 1997
"... A kcritrical graph G has maximum degree k 0, chromatic index Ø 0 (G) = k + 1 and Ø 0 (G \Gamma e) ! k + 1 for each edge e of G. The Critical Graph Conjecture, Jakobsen [8] and Beineke, Wilson [1], claims that every kcritical graph is of odd order. Fiorini and Wilson [6] conjectured that ev ..."
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A kcritrical graph G has maximum degree k 0, chromatic index Ø 0 (G) = k + 1 and Ø 0 (G \Gamma e) ! k + 1 for each edge e of G. The Critical Graph Conjecture, Jakobsen [8] and Beineke, Wilson [1], claims that every kcritical graph is of odd order. Fiorini and Wilson [6] conjectured that every kcritical graph of even order has a 1factor. Chetwynd and Yap [4] stated the problem whether it is true that if G is a kcritical graph of odd order, then G \Gamma v has a 1factor for every vertex v of minimum degree. These conjectures are disproved and the problem is answered in the negative for k 2 f3; 4g. We disprove these conjectures and answer the problem in the negative for all k 3. We also construct kcritical graphs on n vertices with degree sequence 23 2 4 n\Gamma3 , answering a question of Yap [11]. 1 Introduction We consider connected multigraphs M = (V (M); E(M)) without loops, where V (M) (E(M)) denotes the set of vertices (edges) of M . The degree dM (v) of a v...
Supereulerian Graphs and the Petersen Graph
, 2014
"... A graph G is supereulerian if G has a spanning eulerian subgraph. Boesch et al. [J. Graph Theory, 1, 79–84 (1977)] proposed the problem of characterizing supereulerian graphs. In this paper, we prove that any 3edgeconnected graph with at most 11 edgecuts of size 3 is supereulerian if and only if ..."
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A graph G is supereulerian if G has a spanning eulerian subgraph. Boesch et al. [J. Graph Theory, 1, 79–84 (1977)] proposed the problem of characterizing supereulerian graphs. In this paper, we prove that any 3edgeconnected graph with at most 11 edgecuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph. This extends a former result of Catlin and Lai [J. Combin. Theory, Ser. B, 66, 123–139 (1996)].