Results 11  20
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1,271
Irregularity Strength of Dense Graphs
, 2008
"... Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f: E(G) → {1, 2,..., w}. For a vertex v, f(v) is defined as the sum f(e) over all edges e of G incident with v. f is called irregular, if all f(v) are dist ..."
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Cited by 3 (0 self)
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the argument for dregular graphs with d ≥ 10 4/3 n 2/3 log 1/3 n, we prove that s(G) ≤ 48(n/d) + 6.
Property Testing in Bounded Degree Graphs
 Algorithmica
, 1997
"... We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength in ..."
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Cited by 124 (36 self)
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for testing whether an unknown boundeddegree graph is connected, kconnected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is fflaway from having the property. For example
CONNECTED VERTEX COVERS IN DENSE GRAPHS
"... Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parame ..."
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Cited by 7 (1 self)
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parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All
Connectivity of Strong Products of Graphs
"... Definition(s): Let G = (V,E) be a graph. A set S ⊆ V is called separating in G if G − S is not connected. The connectivity of G, written κ(G), is the minimum size of a set S, such that G−S is not connected or has only one vertex. A separating set in G with cardinality κ(G) is called a κset in G. Le ..."
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. Let G1 = (V1, E1) and G2 = (V2, E2) be a graphs. Strong product G1 G2 of graphs G1 and G2 is the graph with V (G1 G2) = V1 × V2, where vertices (x1, x2) and (y1, y2) are adjacent if one of the following occurs • x1 = y1 and x2y2 ∈ E2, • x2 = y2 and x1y1 ∈ E1, • x1y1 ∈ E1 and x2y2 ∈ E2. If a set I
Good and Semistrong Colorings of Oriented Planar Graphs
 INF. PROCESSING LETTERS 51
, 1994
"... A kcoloring of an oriented graph G = (V, A) is an assignment c of one of the colors 1; 2; : : : ; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6= c(y). The kcoloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). ..."
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Cited by 53 (21 self)
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A kcoloring of an oriented graph G = (V, A) is an assignment c of one of the colors 1; 2; : : : ; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6= c(y). The kcoloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z
Endpoint Extendible Paths in Dense Graphs
, 2005
"... Let G be a graph of order n. A path P of G is extendible if it can be extended to a longer path from one of its two endvertices, otherwise we say P is nonextendible. Let G be a graph of order n. We show that there exists a threshold number s such that every path of order smaller than s is extendibl ..."
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s is extendible and there exists a nonextendible path of order t for each t ∈ {s,s + 1, · · ·,n} provided G satisfies one of the following three conditions: • d(u) + d(v) ≥ n for any two of nonadjacent vertices u and v. • G is a P4free 1tough graph. • G is a connected, locally connected, and K1,3free
ON THE ORIENTED CHROMATIC NUMBER OF DENSE GRAPHS
, 2006
"... Abstract. Let G be a graph with n vertices, m edges, average degree δ, and maximum degree ∆. The oriented chromatic number of G is the maximum, taken over all orientations of G, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges hav ..."
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Cited by 1 (0 self)
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connection with harmonious colourings, we prove a general upper bound of O( ∆ √ n) on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when G is (clog n)regular for some constant c> 2, in which case the oriented
On the strong chromatic number of graphs
, 2006
"... The strong chromatic number, χS(G), of an nvertex graph G is the smallest number k such that after adding k⌈n/k ⌉ − n isolated vertices to G and considering any partition of the vertices of the resulting graph into disjoint subsets V1,..., V ⌈n/k ⌉ of size k each, one can find a proper kvertexco ..."
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The strong chromatic number, χS(G), of an nvertex graph G is the smallest number k such that after adding k⌈n/k ⌉ − n isolated vertices to G and considering any partition of the vertices of the resulting graph into disjoint subsets V1,..., V ⌈n/k ⌉ of size k each, one can find a proper k
On quasistrongly regular graphs
, 2004
"... We study the quasistrongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasistrongly regular graphs of grade 2. We prove a ‘‘spectral gap’’type result for them which generalizes Seidel’s wellknown formula ..."
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We study the quasistrongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasistrongly regular graphs of grade 2. We prove a ‘‘spectral gap’’type result for them which generalizes Seidel’s wellknown formula
Strongly connected graphs and polynomials
, 2011
"... In this report, we give the exact solutions ofthe equation P(A) = 0 where P is a polynomial of degree2 with integer coefficients, and A is the adjacency matrix of a strongly connected graph. Then we study the problem for P a polynomial of degree n ≥ 3, and give a necessary condition on the trace of ..."
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In this report, we give the exact solutions ofthe equation P(A) = 0 where P is a polynomial of degree2 with integer coefficients, and A is the adjacency matrix of a strongly connected graph. Then we study the problem for P a polynomial of degree n ≥ 3, and give a necessary condition on the trace
Results 11  20
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1,271