Abstract A graph is k -critical if it is k -chromatic but each of its proper induced subgraphs is... more Abstract A graph is k -critical if it is k -chromatic but each of its proper induced subgraphs is ( k − 1 )-colorable. It is known that the number of 4 -critical P 5 -free graphs is finite, but there is an infinite number of k -critical P 5 -free graphs for each k ≥ 5 . We show that the number of k -critical ( P 5 , P ¯ 5 ) -free graphs is finite for every fixed k . Our result implies the existence of a certifying algorithm for k -coloring ( P 5 , P ¯ 5 ) -free graphs.
Abstract Given a set L of graphs, a graph G is L -free if G does not contain any graph in L as an... more Abstract Given a set L of graphs, a graph G is L -free if G does not contain any graph in L as an induced subgraph. There has been keen interest in colouring graphs whose forbidden list L contains graphs with four vertices. A recent paper of Lozin and Malyshev (in press) discusses the state of the art on this problem, identifying three outstanding classes: L = ( 4 K 1 , claw ) , L = ( 4 K 1 , claw, co-diamond ) , and L = ( 4 K 1 , C 4 ) . In this paper we investigate the class of ( 4 K 1 , C 4 , C 5 )-free graphs and show that if G is a ( 4 K 1 , C 4 , C 5 )-free graph, then G either has bounded clique width or is perfect.
Page 1. On the Two-Edge-Colorings of Perfect Graphs Chinh T. Hoang DEPARTMENT OF MA THEMA TlCA L ... more Page 1. On the Two-Edge-Colorings of Perfect Graphs Chinh T. Hoang DEPARTMENT OF MA THEMA TlCA L SCIENCES LA KEH AD U NIVERSIN THUNDER BAY, ONTARIO, CANADA ABSTRACT We investigate the conjecture ...
In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs co... more In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs and the triangulated graphs. In this paper, we introduce four classes of perfectly orderable graphs, including natural generalizations of the comparability and triangulated graphs. We provide recognition algorithms for these four classes. We also discuss how to solve the clique, clique cover, coloring, and stable set problems for these classes.
Abstract A graph is k -critical if it is k -chromatic but each of its proper induced subgraphs is... more Abstract A graph is k -critical if it is k -chromatic but each of its proper induced subgraphs is ( k − 1 )-colorable. It is known that the number of 4 -critical P 5 -free graphs is finite, but there is an infinite number of k -critical P 5 -free graphs for each k ≥ 5 . We show that the number of k -critical ( P 5 , P ¯ 5 ) -free graphs is finite for every fixed k . Our result implies the existence of a certifying algorithm for k -coloring ( P 5 , P ¯ 5 ) -free graphs.
Abstract Given a set L of graphs, a graph G is L -free if G does not contain any graph in L as an... more Abstract Given a set L of graphs, a graph G is L -free if G does not contain any graph in L as an induced subgraph. There has been keen interest in colouring graphs whose forbidden list L contains graphs with four vertices. A recent paper of Lozin and Malyshev (in press) discusses the state of the art on this problem, identifying three outstanding classes: L = ( 4 K 1 , claw ) , L = ( 4 K 1 , claw, co-diamond ) , and L = ( 4 K 1 , C 4 ) . In this paper we investigate the class of ( 4 K 1 , C 4 , C 5 )-free graphs and show that if G is a ( 4 K 1 , C 4 , C 5 )-free graph, then G either has bounded clique width or is perfect.
Page 1. On the Two-Edge-Colorings of Perfect Graphs Chinh T. Hoang DEPARTMENT OF MA THEMA TlCA L ... more Page 1. On the Two-Edge-Colorings of Perfect Graphs Chinh T. Hoang DEPARTMENT OF MA THEMA TlCA L SCIENCES LA KEH AD U NIVERSIN THUNDER BAY, ONTARIO, CANADA ABSTRACT We investigate the conjecture ...
In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs co... more In 1981, Chvátal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs and the triangulated graphs. In this paper, we introduce four classes of perfectly orderable graphs, including natural generalizations of the comparability and triangulated graphs. We provide recognition algorithms for these four classes. We also discuss how to solve the clique, clique cover, coloring, and stable set problems for these classes.
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