Graph Theory For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G,... more
Graph Theory For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V (G) such that |N [v] ∩ S| ≥ k for every vertex v, where N [v] = {v} ∪ {u ∈ V (G) : uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ×k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ×k (G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ×2 (G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.
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Graphs and Algorithms A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c... more
Graphs and Algorithms A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].
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The induced star arboricity (isa($G$)) of a graph $G$ is a new parameter introduced in 2019 by Axenovich et \textit{al.} \cite{axia2019}, it is defined as the smallest number of induced star-forests covering the edges of $G$. An injective... more
The induced star arboricity (isa($G$)) of a graph $G$ is a new parameter introduced in 2019 by Axenovich et \textit{al.} \cite{axia2019}, it is defined as the smallest number of induced star-forests covering the edges of $G$. An injective edge-coloring $c$ of a graph $G$ is an edge coloring such that if $e_1$, $e_2$ and $e_3$ are three consecutive edges in $G$ (they are consecutive if they form a path or a cycle of length three), then $e_1$ and $e_3$ receive different colors. The minimum integer $k$ such that $G$ has an injective $k$-edge-coloring is called injective chromatic index of $G$ ($\chi'_{inj}(G)$). This parameter was introduced in 2015 by Cardoso et \textit{al.} \cite{CCCD} motivated by the Packet Radio Network problem. They proved that computing $\chi'_{inj}(G)$ of a graph $G$ is NP-hard and they gave the injective chromatic index of some classes of graphs. In our paper we first prove that for any graph $G$ we have isa($G$)=$\chi'_{inj}(G)$. We give some new ...
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In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the... more
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks' theorem to signed graphs.
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ABSTRACT
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A subset of nodes S in a graph G is called k-dominating if, for every node u of the graph, the distance from u to S is at most k. We consider the parameter k (G) dened as the smallest integer r such that G has a k-dominating set of... more
A subset of nodes S in a graph G is called k-dominating if, for every node u of the graph, the distance from u to S is at most k. We consider the parameter k (G) dened as the smallest integer r such that G has a k-dominating set of cardinality r. For planar graphs, we show that for every > 0 and for every k ( 5 7 + )D, k (G) = O(1=). For several classes of planar graphs of diameter D, we show that k (G) is bounded by a constant for k 1 2 D. We conjecture that the same result holds for every planar graph. This problem is motivated by the design of routing schemes with compact data structures.
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The (d; 1)-total number d (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same label, no two incident edges have the same... more
The (d; 1)-total number d (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least d. This notion was introduced in [HY02]. In this paper, we prove that d (G) +2d 2 for connected planar graphs with large girth and high maximum degree . Our results are optimal for d = 2.
A digraph G is k-nice for some positive integer k if for every two (not necessarily distinct) vertices x and y in G and every pattern of length k, given as a sequence of pluses and minuses, there exists a walk of length k linking x to y... more
A digraph G is k-nice for some positive integer k if for every two (not necessarily distinct) vertices x and y in G and every pattern of length k, given as a sequence of pluses and minuses, there exists a walk of length k linking x to y which respects this pattern (pluses corresponding to forward edges and minuses to backward edges). A digraph is then nice if it is k-nice for some k. Similarly, a multigraph This research was supported by the National Sciences and Engineering Research Council of Canada. y Part of this work was done while the author was visiting LaBRI. This author finished his part while visiting Nottingham University, funded by Visiting Fellowship Research Grant GR/L54585 from the Engineering and Physical Sciences Research Council. His research was also supported in part by the grant 97-01-01075 of the Russian Foundation for Fundamental Research. z Part of this work was done while the author was visiting Simon Fraser University. This research was partly supported...
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A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank kk , k \geqslant 4k \geqslant 4 , in a map on a surface SNS^N... more
A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank kk , k \geqslant 4k \geqslant 4 , in a map on a surface SNS^N is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly on N and kk . In the present paper we prove a sharper estimate 55( - Nk)4/755( - Nk)^{4/7} for the number of colors provided that k \geqslant 1k \geqslant 1 and - N \geqslant 57 k4/3 - N \geqslant 5^7 k^{4/3} .
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Research Interests: Mathematics and Inverse
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ABSTRACT We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with... more
ABSTRACT We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere-zero integer flow. In the present article, we establish the existence of a universal coefficient such that every signed graph G that admits a nowhere-zero integer flow has a signed circuit cover of total length at most . We show that if G is bridgeless, then , and in the general case .
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