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Tytuł:
Nowhere-zero modular edge-graceful graphs
Autorzy:
Jones, Ryan
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743248.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
modular edge-graceful labelings and graphs
nowhere-zero labelings
modular edge-gracefulness
Opis:
For a connected graph G of order n ≥ 3, let f: E(G) → ℤₙ be an edge labeling of G. The vertex labeling f': V(G) → ℤₙ induced by f is defined as $f'(u) = ∑_{v ∈ N(u)} f(uv)$, where the sum is computed in ℤₙ. If f' is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) ≠ 0 for all e ∈ E(G) and in this case, G is a nowhere-zero modular edge-graceful graph. It is shown that a connected graph G of order n ≥ 3 is nowhere-zero modular edge-graceful if and only if n ≢ 2 mod 4, G ≠ K₃ and G is not a star of even order. For a connected graph G of order n ≥ 3, the smallest integer k ≥ n for which there exists an edge labeling f: E(G) → ℤₖ - {0} such that the induced vertex labeling f' is one-to-one is referred to as the nowhere-zero modular edge-gracefulness of G and this number is determined for every connected graph of order at least 3.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 3; 487-505
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Connected partition dimensions of graphs
Autorzy:
Saenpholphat, Varaporn
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743364.pdf
Data publikacji:
2002
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
distance
resolving partition
connected resolving partition
Opis:
For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = min{d(v,x)|x ∈ S}. For an ordered k-partition Π = {S₁,S₂,...,Sₖ} of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S₁), d(v,S₂),..., d(v,Sₖ)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = {S₁,S₂,...,Sₖ} of V(G) is connected if each subgraph $⟨S_i⟩$ induced by $S_i$ (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V(G) is the connected partition dimension cpd(G) of G. Thus 2 ≤ pd (G) ≤ cpd(G) ≤ n for every connected graph G of order n ≥ 2. The connected partition dimensions of several classes of well-known graphs are determined. It is shown that for every pair a, b of integers with 3 ≤ a ≤ b ≤ 2a-1, there is a connected graph G having pd(G) = a and cpd(G) = b. Connected graphs of order n ≥ 3 having connected partition dimension 2, n, or n-1 are characterized.
Źródło:
Discussiones Mathematicae Graph Theory; 2002, 22, 2; 305-323
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On graphs with a unique minimum hull set
Autorzy:
Chartrand, Gary
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743417.pdf
Data publikacji:
2001
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
geodetic set
geodetic number
convex hull
hull set
hull number
hull graph
Opis:
We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L(v_i) = G_i$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2001, 21, 1; 31-42
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On stratification and domination in graphs
Autorzy:
Gera, Ralucca
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743948.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
stratified graph
F-domination
domination
Opis:
A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F-coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F-domination number $γ_F(G)$ is the minimum number of red vertices in an F-coloring of G. In this paper, we study F-domination, where F is a 2-stratified red-blue-blue path of order 3 rooted at a blue end-vertex. We present characterizations of connected graphs of order n with F-domination number n or 1 and establish several realization results on F-domination number and other domination parameters.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 2; 249-272
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
The forcing geodetic number of a graph
Autorzy:
Chartrand, Gary
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/744241.pdf
Data publikacji:
1999
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
geodetic set
geodetic number
forcing geodetic number
Opis:
For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u-v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u,v) for u, v ∈ S. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number $f_G(S)$ of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a,b) ∉ {(1,1),(2,2)}.
Źródło:
Discussiones Mathematicae Graph Theory; 1999, 19, 1; 45-58
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Closed Modular Colorings of Trees
Autorzy:
Phinezy, Bryan
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/30146543.pdf
Data publikacji:
2013-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
trees
closed modular k-coloring
closed modular chromatic number
Opis:
Two vertices $u$ and $v$ in a nontrivial connected graph $G$ are twins if $u$ and $v$ have the same neighbors in $V (G)$ − ${u, v}$. If $u$ and $v$ are adjacent, they are referred to as true twins; while if $u$ and $v$ are nonadjacent, they are false twins. For a positive integer $k$, let $c : V (G) \rightarrow \mathbb{Z}_k $ be a vertex coloring where adjacent vertices may be assigned the same color. The coloring $c$ induces another vertex coloring $ c^′ : V (G) \rightarrow \mathbb{Z}_k $ defined by $ c′(v) = \Sigma_{u \in N[v]} c(u) $ for each $ v \in V (G) $, where $N[v]$ is the closed neighborhood of $v$. Then $c$ is called a closed modular $k$-coloring if $c^′(u) \ne c′(v)$ in $ \mathbb{Z}_k$ for all pairs $u$, $v$ of adjacent vertices that are not true twins. The minimum $k$ for which $G$ has a closed modular $k$-coloring is the closed modular chromatic number $ \overline{mc}(G) $ of $G$. The closed modular chromatic number is investigated for trees and determined for several classes of trees. For each tree $T$ in these classes, it is shown that $ \overline{mc} (T) = 2$ or $ \overline{mc}(T) = 3 $. A closed modular $k$-coloring $c$ of a tree $T$ is called nowhere-zero if $c(x) \ne 0 $ for each vertex $x$ of $T$. It is shown that every tree of order 3 or more has a nowhere-zero closed modular 4-coloring.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 2; 411-428
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On k-Path Pancyclic Graphs
Autorzy:
Bi, Zhenming
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/31339488.pdf
Data publikacji:
2015-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Hamiltonian
panconnected
pancyclic
path Hamiltonian
path pancyclic
Opis:
For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 2; 271-281
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A note on total colorings of planar graphs without 4-cycles
Autorzy:
Wang, Ping
Wu, Jian-Liang
Powiązania:
https://bibliotekanauki.pl/articles/744436.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total coloring
planar graph
list coloring
girth
Opis:
Let G be a 2-connected planar graph with maximum degree Δ such that G has no cycle of length from 4 to k, where k ≥ 4. Then the total chromatic number of G is Δ +1 if (Δ,k) ∈ {(7,4),(6,5),(5,7),(4,14)}.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 1; 125-135
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Distance defined by spanning trees in graphs
Autorzy:
Chartrand, Gary
Nebeský, Ladislav
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743431.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
distance
geodesic
T-path
T-geodesic
T-distance
Opis:
For a spanning tree T in a nontrivial connected graph G and for vertices u and v in G, there exists a unique u-v path u = u₀, u₁, u₂,..., uₖ = v in T. A u-v T-path in G is a u- v path u = v₀, v₁,...,vₗ = v in G that is a subsequence of the sequence u = u₀, u₁, u₂,..., uₖ = v. A u-v T-path of minimum length is a u-v T-geodesic in G. The T-distance $d_{G|T}(u,v)$ from u to v in G is the length of a u-v T-geodesic. Let geo(G) and geo(G|T) be the set of geodesics and the set of T-geodesics respectively in G. Necessary and sufficient conditions are established for (1) geo(G) = geo(G|T) and (2) geo(G|T) = geo(G|T*), where T and T* are two spanning trees of G. It is shown for a connected graph G that geo(G|T) = geo(G) for every spanning tree T of G if and only if G is a block graph. For a spanning tree T of a connected graph G, it is also shown that geo(G|T) satisfies seven of the eight axioms of the characterization of geo(G). Furthermore, we study the relationship between the distance d and T-distance $d_{G|T}$ in graphs and present several realization results on parameters and subgraphs defined by these two distances.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 3; 485-506
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Characterizations of Graphs Having Large Proper Connection Numbers
Autorzy:
Lumduanhom, Chira
Laforge, Elliot
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/31340917.pdf
Data publikacji:
2016-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
edge coloring
proper-path coloring
strong proper-path coloring
Opis:
Let G be an edge-colored connected graph. A path P is a proper path in G if no two adjacent edges of P are colored the same. If P is a proper u − v path of length d(u, v), then P is a proper u − v geodesic. An edge coloring c is a proper-path coloring of a connected graph G if every pair u, v of distinct vertices of G are connected by a proper u − v path in G, and c is a strong proper-path coloring if every two vertices u and v are connected by a proper u− v geodesic in G. The minimum number of colors required for a proper-path coloring or strong proper-path coloring of G is called the proper connection number pc(G) or strong proper connection number spc(G) of G, respectively. If G is a nontrivial connected graph of size m, then pc(G) ≤ spc(G) ≤ m and pc(G) = m or spc(G) = m if and only if G is the star of size m. In this paper, we determine all connected graphs G of size m for which pc(G) or spc(G) is m − 1,m − 2 or m − 3.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 2; 439-453
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Eulerian irregularity in graphs
Autorzy:
Andrews, Eric
Lumduanhom, Chira
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/31232740.pdf
Data publikacji:
2014-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Eulerian walks
Eulerian irregularity
Opis:
A closed walk in a connected graph $G$ that contains every edge of $G$ exactly once is an Eulerian circuit. A graph is Eulerian if it contains an Eulerian circuit. It is well known that a connected graph $G$ is Eulerian if and only if every vertex of $G$ is even. An Eulerian walk in a connected graph $G$ is a closed walk that contains every edge of $G$ at least once, while an irregular Eulerian walk in $G$ is an Eulerian walk that encounters no two edges of $G$ the same number of times. The minimum length of an irregular Eulerian walk in $G$ is called the Eulerian irregularity of $G$ and is denoted by $EI(G)$. It is known that if $G$ is a nontrivial connected graph of size $m$, then \(\binom{m+1}{2} \le EI(G) \le 2 \binom{m+1}{2}\). A necessary and sufficient condition has been established for all pairs $k, m$ of positive integers for which there is a nontrivial connected graph $G$ of size $m$ with $EI(G)=k$. A subgraph $F$ in a graph $G$ is an even subgraph of $G$ if every vertex of $F$ is even. We present a formula for the Eulerian irregularity of a graph in terms of the size of certain even subgraph of the graph. Furthermore, Eulerian irregularities are determined for all graphs of cycle rank 2 and all complete bipartite graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2014, 34, 2; 391-408
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Radio k-colorings of paths
Autorzy:
Chartrand, Gary
Nebeský, Ladislav
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743194.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
radio k-coloring
radio k-chromatic number
Opis:
For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that
d(u,v) + |c(u)- c(v)| ≥ 1 + k
for every two distinct vertices u and v of G, where d(u,v) is the distance between u and v. The value rcₖ(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rcₖ(G) of G is the minimum value of rcₖ(c) taken over all radio k-colorings c of G. In this paper, radio k-colorings of paths are studied. For the path Pₙ of order n ≥ 9 and n odd, a new improved bound for $rc_{n- 2}(Pₙ)$ is presented. For n ≥ 4, it is shown that
$rc_{n-3}(Pₙ) ≤ \binom{n-2} {2}$
Upper and lower bounds are also presented for rcₖ(Pₙ) in terms of k when 1 ≤ k ≤ n- 1. The upper bound is shown to be sharp when 1 ≤ k ≤ 4 and n is sufficiently large.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 1; 5-21
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Geodetic sets in graphs
Autorzy:
Chartrand, Gary
Harary, Frank
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743733.pdf
Data publikacji:
2000
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
geodetic set
geodetic number
upper geodetic number
Opis:
For two vertices u and v of a graph G, the closed interval I[u,v] consists of u, v, and all vertices lying in some u-v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u,v] for u, v ∈ S. If I[S] = V(G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for every two distinct vertices u,v ∈ S, there exists a third vertex w of G that lies in some u-v geodesic but in no x-y geodesic for x, y ∈ S and {x,y} ≠ {u,v}. It is shown that for every integer k ≥ 2, there exists a connected graph G with g(G) = k which contains a uniform, essential minimum geodetic set. A minimal geodetic set S has no proper subset which is a geodetic set. The maximum cardinality of a minimal geodetic set is the upper geodetic number g⁺(G). It is shown that every two integers a and b with 2 ≤ a ≤ b are realizable as the geodetic and upper geodetic numbers, respectively, of some graph and when a < b the minimum order of such a graph is b+2.
Źródło:
Discussiones Mathematicae Graph Theory; 2000, 20, 1; 129-138
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On multiset colorings of graphs
Autorzy:
Okamoto, Futaba
Salehi, Ebrahim
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/744555.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
vertex coloring
multiset coloring
neighbor-distinguishing coloring
Opis:
A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χₘ(G) of G. For every graph G, χₘ(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that for every pair k, r of integers with 2 ≤ k ≤ r - 1, there exists an r-regular graph with multiset chromatic number k. It is also shown that for every positive integer N, there is an r-regular graph G such that χ(G) - χₘ(G) = N. In particular, it is shown that χₘ(Kₙ × K₂) is asymptotically √n. In fact, $χₘ(Kₙ × K₂) = χₘ(cor(K_{n+1}))$. The corona cor(G) of a graph G is the graph obtained from G by adding, for each vertex v in G, a new vertex v' and the edge vv'. It is shown that χₘ(cor(G)) ≤ χₘ(G) for every nontrivial connected graph G. The multiset chromatic numbers of the corona of all complete graphs are determined. On Multiset Colorings of Graphs From this, it follows that for every positive integer N, there exists a graph G such that χₘ(G) - χₘ(cor(G)) ≥ N. The result obtained on the multiset chromatic number of the corona of complete graphs is then extended to the corona of all regular complete multipartite graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 1; 137-153
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Kaleidoscopic Colorings of Graphs
Autorzy:
Chartrand, Gary
English, Sean
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/31341692.pdf
Data publikacji:
2017-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
edge coloring
vertex coloring
kaleidoscopic coloring
kaleidoscope
Opis:
For an $r$-regular graph $G$, let $ c : E(G) \rightarrow [k] = {1, 2, . . ., k}$, $ k \ge 3 $, be an edge coloring of $G$, where every vertex of $G$ is incident with at least one edge of each color. For a vertex $v$ of $G$, the multiset-color $ c_m(v)$ of $v$ is defined as the ordered $k$-tuple $ (a_1, a_2, . . ., a_k) $ or $ a_1a_2…a_k$, where $a_i$ $(1 \le i \le k)$ is the number of edges in $G$ colored $i$ that are incident with $v$. The edge coloring $c$ is called $k$-kaleidoscopic if $ c_m(u) \ne c_m(v)$ for every two distinct vertices $u$ and $v$ of $G$. A regular graph $G$ is called a $k$-kaleidoscope if $G$ has a $k$-kaleidoscopic coloring. It is shown that for each integer $k \ge 3 $, the complete graph $ K_{k+3}$ is a $k$-kaleidoscope and the complete graph $ K_n $ is a 3-kaleidoscope for each integer $ n \ge 6 $. The largest order of an $r$-regular 3-kaleidoscope is \( \binom{r-1}{2} \). It is shown that for each integer $ r \ge 5 $ such that \( r \not\equiv 3 (\text{mod } 4) \), there exists an $r$-regular 3-kaleidoscope of order \( \binom{r-1}{2} \).
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 3; 711-727
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 27

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