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Papers by Leonardo Sampaio

arXiv (Cornell University), Jul 24, 2018

A connected ordering (v1, v2,. .. , vn) of V (G) is an ordering of the vertices such that vi has ... more A connected ordering (v1, v2,. .. , vn) of V (G) is an ordering of the vertices such that vi has at least one neighbour in {v1,. .. , vi−1} for every i ∈ {2,. .. , n}. A connected greedy coloring (CGC for short) is a coloring obtained by applying the greedy algorithm to a connected ordering. This has been first introduced in 1989 by Hertz and de Werra, but still very little is known about this problem. An interesting aspect is that, contrary to the traditional greedy coloring, it is not always true that a graph has a connected ordering that produces an optimal coloring; this motivates the definition of the connected chromatic number of G, which is the smallest value χc(G) such that there exists a CGC of G with χc(G) colors. An even more interesting fact is that χc(G) ≤ χ(G) + 1 for every graph G (Benevides et. al. 2014). In this paper, in the light of the dichotomy for the coloring problem restricted to H-free graphs given by Král et.al. in 2001, we are interested in investigating the problems of, given an H-free graph G: (1). deciding whether χc(G) = χ(G); and (2). given also a positive integer k, deciding whether χc(G) ≤ k. We have proved that Problem (2) has the same dichotomy as the coloring problem (i.e., it is polynomial when H is an induced subgraph of P4 or of P3 + K1, and it is NPcomplete otherwise). As for Problem (1), we have proved that χc(G) = χ(G) always hold when G is an induced subgraph of P5 or of P4 + K1, and that it is NP-complete to decide whether χc(G) = χ(G) when H is not a linear forest or contains an induced P9. We mention that some of the results actually involve fixed k and fixed χ(G).

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a ve... more A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to a vertex in each other color class. The bchromatic number of G is the maximum integer χb(G) for which G has a b-coloring with χb(G) colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing χb(G )i s NP -hard in general and polynomialtime solvable for trees. A natural question that arises is whether the edge version of this problem is also NP -hard or not. Here, we prove that computing the bchromatic index of a graph G is NP -hard, even if G is either a comparability graph or a Ck-free graph, and give some partial results on the complexity of the problem restricted to trees.

Lecture Notes in Computer Science, 2014

ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 <v 2 <⋯&lt... more ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 <v 2 <⋯<v n of V(G) such that v i has at least one neighbour in {v 1 ,⋯,v i-1 }, for every i∈{2,⋯,n}. A connected greedy colouring is a colouring obtained by the greedy algorithm applied to a connected vertex ordering. In this paper we study the parameter Γ c (G), which is the maximum k such that G admits a connected greedy k-colouring, and χ c (G), which is the minimum k such that a connected greedy k-colouring of G exists. We prove that computing Γ c (G) is NP-hard for chordal graphs and complements of bipartite graphs. We also prove that if G is bipartite, Γ c (G)=2. Concerning χ c (G), we first show that there is a k-chromatic graph G k with χ c (G k )>χ(G k ), for every k≥3. We then prove that for every graph G,χ c (G)≤χ(G)+1. Finally, we prove that deciding if χ c (G)=χ(G), given a graph G, is a NP-hard problem.

Electronic Notes in Discrete Mathematics, 2013

ABSTRACT A b-coloring of the vertices of a graph is a proper coloring where each color class cont... more ABSTRACT A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of G is the maximum integer b(G) for which G has a b-coloring with b(G) colors. This problem was introduced by Irving and Manlove (1999), where they showed that computing b(G) is NP-hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also NP-hard or not. Here, we prove that computing the b-chromatic index of a graph G is NP-hard, even if G is either a comparability graph or a Ck-free graph, and give partial results on the complexity of the problem restricted to trees, more specifically, we solve the problem for caterpillars graphs. Although solving problems on caterpillar graphs is usually quite simple, this problem revealed itself to be unusually hard. The presented algorithm uses a dynamic programming approach that combines partial solutions which are proved to exist if, and only if, a particular polyhedron is non-empty.

arXiv (Cornell University), Jul 24, 2018

A connected ordering (v1, v2,. .. , vn) of V (G) is an ordering of the vertices such that vi has ... more A connected ordering (v1, v2,. .. , vn) of V (G) is an ordering of the vertices such that vi has at least one neighbour in {v1,. .. , vi−1} for every i ∈ {2,. .. , n}. A connected greedy coloring (CGC for short) is a coloring obtained by applying the greedy algorithm to a connected ordering. This has been first introduced in 1989 by Hertz and de Werra, but still very little is known about this problem. An interesting aspect is that, contrary to the traditional greedy coloring, it is not always true that a graph has a connected ordering that produces an optimal coloring; this motivates the definition of the connected chromatic number of G, which is the smallest value χc(G) such that there exists a CGC of G with χc(G) colors. An even more interesting fact is that χc(G) ≤ χ(G) + 1 for every graph G (Benevides et. al. 2014). In this paper, in the light of the dichotomy for the coloring problem restricted to H-free graphs given by Král et.al. in 2001, we are interested in investigating the problems of, given an H-free graph G: (1). deciding whether χc(G) = χ(G); and (2). given also a positive integer k, deciding whether χc(G) ≤ k. We have proved that Problem (2) has the same dichotomy as the coloring problem (i.e., it is polynomial when H is an induced subgraph of P4 or of P3 + K1, and it is NPcomplete otherwise). As for Problem (1), we have proved that χc(G) = χ(G) always hold when G is an induced subgraph of P5 or of P4 + K1, and that it is NP-complete to decide whether χc(G) = χ(G) when H is not a linear forest or contains an induced P9. We mention that some of the results actually involve fixed k and fixed χ(G).

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a ve... more A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to a vertex in each other color class. The bchromatic number of G is the maximum integer χb(G) for which G has a b-coloring with χb(G) colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing χb(G )i s NP -hard in general and polynomialtime solvable for trees. A natural question that arises is whether the edge version of this problem is also NP -hard or not. Here, we prove that computing the bchromatic index of a graph G is NP -hard, even if G is either a comparability graph or a Ck-free graph, and give some partial results on the complexity of the problem restricted to trees.

Lecture Notes in Computer Science, 2014

ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 <v 2 <⋯&lt... more ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 <v 2 <⋯<v n of V(G) such that v i has at least one neighbour in {v 1 ,⋯,v i-1 }, for every i∈{2,⋯,n}. A connected greedy colouring is a colouring obtained by the greedy algorithm applied to a connected vertex ordering. In this paper we study the parameter Γ c (G), which is the maximum k such that G admits a connected greedy k-colouring, and χ c (G), which is the minimum k such that a connected greedy k-colouring of G exists. We prove that computing Γ c (G) is NP-hard for chordal graphs and complements of bipartite graphs. We also prove that if G is bipartite, Γ c (G)=2. Concerning χ c (G), we first show that there is a k-chromatic graph G k with χ c (G k )>χ(G k ), for every k≥3. We then prove that for every graph G,χ c (G)≤χ(G)+1. Finally, we prove that deciding if χ c (G)=χ(G), given a graph G, is a NP-hard problem.

Electronic Notes in Discrete Mathematics, 2013

ABSTRACT A b-coloring of the vertices of a graph is a proper coloring where each color class cont... more ABSTRACT A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of G is the maximum integer b(G) for which G has a b-coloring with b(G) colors. This problem was introduced by Irving and Manlove (1999), where they showed that computing b(G) is NP-hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also NP-hard or not. Here, we prove that computing the b-chromatic index of a graph G is NP-hard, even if G is either a comparability graph or a Ck-free graph, and give partial results on the complexity of the problem restricted to trees, more specifically, we solve the problem for caterpillars graphs. Although solving problems on caterpillar graphs is usually quite simple, this problem revealed itself to be unusually hard. The presented algorithm uses a dynamic programming approach that combines partial solutions which are proved to exist if, and only if, a particular polyhedron is non-empty.