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The bchromatic number of cubic graphs
"... The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the bchromatic number of cubic graphs is 4. The ..."
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The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the bchromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K3,3, the prism over K3, and one more sporadic example on 10 vertices.
On the bchromatic number of regular graphs ∗
, 2010
"... The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every dregular graph with at least 2d3 vertices has bchromatic number d+ ..."
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Cited by 1 (1 self)
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The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every dregular graph with at least 2d3 vertices has bchromatic number d+1, that the bchromatic number of an arbitrary dregular graph with girth g = 5 is at least ⌊ ⌋ d+1 2 and that every dregular graph, d ≥ 6, with diameter at least d and with no 4cycles admits a bcoloring with d+1 colors.
On the bicoloring of cographs and . . .
"... A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G ..."
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A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G is bcontinuous if it admits a bcoloring with t colors, for every t = χ(G),..., χb(G). We define a graph G to be bmonotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. In this work, we prove that P4sparse graphs (and, in particular, cographs) are bcontinuous and bmonotonic. Besides, we describe a dynamic programming algorithm to compute the bchromatic number in polynomial time within these graph classes.
On the bcoloring of P4tidy graphs
"... A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G i ..."
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A bcoloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The bchromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a bcoloring with t colors. A graph G is bcontinuous if it admits a bcoloring with t colors, for every t = χ(G),..., χb(G), and it is bmonotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. In this work, we prove that P4tidy graphs (a generalization of many classes of graphs with few induced P4s) are bcontinuous and bmonotonic. Furthermore, we describe a polynomial time algorithm to compute the bchromatic number for this class of graphs. Key words: bcoloring, bcontinuity, bmonotonicity, P4tidy graphs
The bchromatic index of a graph
, 2012
"... The bchromatic index ϕ ′ (G) of a graph G is the largest integer k such that G admits a proper kedge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The bchromatic index of trees is determined and equals either to a natural uppe ..."
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The bchromatic index ϕ ′ (G) of a graph G is the largest integer k such that G admits a proper kedge coloring in which every color class contains at least one edge incident to some edge in all the other color classes. The bchromatic index of trees is determined and equals either to a natural upper bound m ′ (T) or one less, where m ′ (T) is connected with the number of edges of high degree. Some conditions are given for which graphs have the bchromatic index strictly less than m ′ (G), and for which conditions it is exactly m ′ (G). In the last part of the paper regular graphs are considered. It is proved that with four exceptions, the bchromatic number of cubic graphs is 5. The exceptions are K4, K3,3, the prism over K3, and the cube Q3. Key words: bchromatic index, regular graphs; AMS subject classification (2010): 05C15, 05C76 1 Introduction and
On the bchromatic number of regular graphs ∗
, 2010
"... The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every dregular graph with at least 2d3 vertices has bchromatic number d+ ..."
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The bchromatic number of a graph G is the largest integer k such that G admits a proper kcoloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every dregular graph with at least 2d3 vertices has bchromatic number d+1, that the bchromatic number of an arbitrary dregular graph with girth g = 5 is at least ⌊ ⌋ d+1 2 and that every dregular graph, d ≥ 6, with diameter at least d and with no 4cycles admits a bcoloring with d+1 colors.