The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-colori... more The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring 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 d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.
ABSTRACT Let G=(V,E)G=(V,E) be a connected graph. The distance between two vertices u,v∈Vu,v∈V, d... more ABSTRACT Let G=(V,E)G=(V,E) be a connected graph. The distance between two vertices u,v∈Vu,v∈V, denoted by d(u,v)d(u,v), is the length of a shortest u,vu,v-path in GG. The distance between a vertex v∈Vv∈V and a subset P⊂VP⊂V is defined as min{d(v,x):x∈P}min{d(v,x):x∈P}, and it is denoted by d(v,P)d(v,P). An ordered partition {P1,P2,…,Pt}{P1,P2,…,Pt} of vertices of a graph GG, is a resolving partition of GG, if all the distance vectors (d(v,P1),d(v,P2),…,d(v,Pt))(d(v,P1),d(v,P2),…,d(v,Pt)) are different. The partition dimension of GG is the minimum number of sets in any resolving partition of GG. In this article we study the partition dimension of strong product graphs and Cartesian product graphs. Specifically, we prove that the partition dimension of the strong product of graphs is bounded below by four and above by the product of the partition dimensions of the factor graphs. Also, we give the exact value of the partition dimension of strong product graphs when one factor is a complete graph and the other one is a path or a cycle. For the case of Cartesian product graphs, we show that its partition dimension is less than or equal to the sum of the partition dimensions of the factor graphs minus one. Moreover, we obtain an upper bound on the partition dimension of Cartesian product graphs, when one factor is a complete graph.
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-colori... more The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring 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 b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K 3,3 , the prism over K 3 , and one more sporadic example on 10 vertices.
A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G... more A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψ k (G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for ψ k of the Cartesian and the direct product of paths are derived. It is shown that for ψ 3 those bounds are tight. For the lexicographic product bounds are presented for ψ k , moreover ψ 2 and ψ 3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.
Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs S n , the Si... more Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs S n , the Sierpiński graphs S(n, k), graphs S + (n, k), and graphs S ++ (n, k) are considered. In particular, χ ′′ (S n ), χ ′ (S(n, k)), χ(S + (n, k)), χ(S ++ (n, k)), χ ′ (S + (n, k)), and χ ′ (S ++ (n, k)) are determined.
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-colori... more The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring 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 d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.
For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G, where adjacent v... more For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G, where adjacent vertices may be colored the same. For a vertex v of G, let N (v) denote the set of vertices adjacent to v. The color sum σ (v) of v is the sum of the colors of the vertices in N (v). If σ (u) = σ (v) for every two adjacent vertices u and v of G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number σ (G). The sigma chromatic number of a graph G never exceeds its chromatic number χ(G) and for every pair a, b of positive integers with a ≤ b, there exists a connected graph G with σ (G) = a and χ(G) = b. There is a connected graph G of order n with σ (G) = k for every pair k, n of positive integers with k ≤ n if and only if k = n − 1. Several other results concerning sigma chromatic numbers are presented.
A subset of vertices of a graph is called a -path vertex cover if every path of order in contains... more A subset of vertices of a graph is called a -path vertex cover if every path of order in contains at least one vertex from . Denote by ( ) the minimum cardinality of a -path vertex cover in . In this paper present an upper bound for 3 of graphs with given average degree. We also give a lower bound for of regular graphs. For the Cartesian products of two paths we give an asymptotically tight bound for and the exact value for 3 .
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-colori... more The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring 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 d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.
ABSTRACT Let G=(V,E)G=(V,E) be a connected graph. The distance between two vertices u,v∈Vu,v∈V, d... more ABSTRACT Let G=(V,E)G=(V,E) be a connected graph. The distance between two vertices u,v∈Vu,v∈V, denoted by d(u,v)d(u,v), is the length of a shortest u,vu,v-path in GG. The distance between a vertex v∈Vv∈V and a subset P⊂VP⊂V is defined as min{d(v,x):x∈P}min{d(v,x):x∈P}, and it is denoted by d(v,P)d(v,P). An ordered partition {P1,P2,…,Pt}{P1,P2,…,Pt} of vertices of a graph GG, is a resolving partition of GG, if all the distance vectors (d(v,P1),d(v,P2),…,d(v,Pt))(d(v,P1),d(v,P2),…,d(v,Pt)) are different. The partition dimension of GG is the minimum number of sets in any resolving partition of GG. In this article we study the partition dimension of strong product graphs and Cartesian product graphs. Specifically, we prove that the partition dimension of the strong product of graphs is bounded below by four and above by the product of the partition dimensions of the factor graphs. Also, we give the exact value of the partition dimension of strong product graphs when one factor is a complete graph and the other one is a path or a cycle. For the case of Cartesian product graphs, we show that its partition dimension is less than or equal to the sum of the partition dimensions of the factor graphs minus one. Moreover, we obtain an upper bound on the partition dimension of Cartesian product graphs, when one factor is a complete graph.
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-colori... more The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring 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 b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K 3,3 , the prism over K 3 , and one more sporadic example on 10 vertices.
A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G... more A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψ k (G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for ψ k of the Cartesian and the direct product of paths are derived. It is shown that for ψ 3 those bounds are tight. For the lexicographic product bounds are presented for ψ k , moreover ψ 2 and ψ 3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.
Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs S n , the Si... more Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs S n , the Sierpiński graphs S(n, k), graphs S + (n, k), and graphs S ++ (n, k) are considered. In particular, χ ′′ (S n ), χ ′ (S(n, k)), χ(S + (n, k)), χ(S ++ (n, k)), χ ′ (S + (n, k)), and χ ′ (S ++ (n, k)) are determined.
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-colori... more The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring 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 d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.
For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G, where adjacent v... more For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G, where adjacent vertices may be colored the same. For a vertex v of G, let N (v) denote the set of vertices adjacent to v. The color sum σ (v) of v is the sum of the colors of the vertices in N (v). If σ (u) = σ (v) for every two adjacent vertices u and v of G, then c is called a sigma coloring of G. The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number σ (G). The sigma chromatic number of a graph G never exceeds its chromatic number χ(G) and for every pair a, b of positive integers with a ≤ b, there exists a connected graph G with σ (G) = a and χ(G) = b. There is a connected graph G of order n with σ (G) = k for every pair k, n of positive integers with k ≤ n if and only if k = n − 1. Several other results concerning sigma chromatic numbers are presented.
A subset of vertices of a graph is called a -path vertex cover if every path of order in contains... more A subset of vertices of a graph is called a -path vertex cover if every path of order in contains at least one vertex from . Denote by ( ) the minimum cardinality of a -path vertex cover in . In this paper present an upper bound for 3 of graphs with given average degree. We also give a lower bound for of regular graphs. For the Cartesian products of two paths we give an asymptotically tight bound for and the exact value for 3 .
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