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Tytuł:
A Note on Neighbor Expanded Sum Distinguishing Index
Autorzy:
Flandrin, Evelyne
Li, Hao
Marczyk, Antoni
Saclé, Jean-François
Woźniak, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/31342189.pdf
Data publikacji:
2017-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
general edge coloring
total coloring
neighbor-distinguishing index
neighbor sum distinguishing coloring
Opis:
A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . ., k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 1; 29-37
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Note on the Total Detection Numbers of Cycles
Autorzy:
Escuadro, Henry E.
Fujie, Futaba
Musick, Chad E.
Powiązania:
https://bibliotekanauki.pl/articles/31339492.pdf
Data publikacji:
2015-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
vertex-distinguishing coloring
detectable labeling
detection number
total detection number
Hamiltonian graph
Opis:
Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . ., k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . ., ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of a detectable labeling c of a graph G is the sum of the labels assigned to the edges in G. The total detection number td(G) of G is defined by td(G) = min{val(c)}, where the minimum is taken over all detectable labelings c of G. We investigate the problem of determining the total detection numbers of cycles.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 2; 237-247
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:
An Improved Upper Bound on Neighbor Expanded Sum Distinguishing Index
Autorzy:
Vučković, Bojan
Powiązania:
https://bibliotekanauki.pl/articles/32083737.pdf
Data publikacji:
2020-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
general edge coloring
total coloring
neighbor sum distinguishing index
Opis:
A total k-weighting f of a graph G is an assignment of integers from the set {1, . . ., k} to the vertices and edges of G. We say that f is neighbor expanded sum distinguishing, or NESD for short, if Σw∈N(v) (f(vw) + f(w)) differs from Σw∈N(u)(f(uw) + f(w)) for every two adjacent vertices v and u of G. The neighbor expanded sum distinguishing index of G, denoted by egndiΣ(G), is the minimum positive integer k for which there exists an NESD weighting of G. An NESD weighting was introduced and investigated by Flandrin et al. (2017), where they conjectured that egndiΣ(G) ≤ 2 for any graph G. They examined some special classes of graphs, while proving that egndiΣ(G) ≤ χ(G) + 1. We improve this bound and show that egndiΣ(G) ≤ 3 for any graph G. We also show that the conjecture holds for all bipartite, 3-regular and 4-regular graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 1; 323-329
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Cubic Graphs with Total Domatic Number at Least Two
Autorzy:
Akbari, Saieed
Motiei, Mohammad
Mozaffari, Sahand
Yazdanbod, Sina
Powiązania:
https://bibliotekanauki.pl/articles/31342441.pdf
Data publikacji:
2018-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
total domatic number
coupon coloring
Opis:
Let G be a graph with no isolated vertex. A total dominating set of G is a set S of vertices of G such that every vertex is adjacent to at least one vertex in S. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of G. In this paper we provide a criterion under which a cubic graph has total domatic number at least two.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 1; 75-82
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles
Autorzy:
Cranston, Daniel
Powiązania:
https://bibliotekanauki.pl/articles/743133.pdf
Data publikacji:
2009
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
list coloring
edge coloring
total coloring
Vizing's Conjecture
Opis:
Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.
Źródło:
Discussiones Mathematicae Graph Theory; 2009, 29, 1; 163-178
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Equitable Total Coloring of Corona of Cubic Graphs
Autorzy:
Furmańczyk, Hanna
Zuazua, Rita
Powiązania:
https://bibliotekanauki.pl/articles/32361758.pdf
Data publikacji:
2021-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
equitable coloring
total coloring
equitable total coloring
cubic graphs
Opis:
The minimum number of total independent partition sets of V∪E of a graph G = (V, E) is called the total chromatic number of G, denoted by χ'′(G). If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V∪E is called the equitable total chromatic number, and is denoted by χ'′=(G). In this paper we consider equitable total coloring of coronas of cubic graphs, G◦H. It turns out that independently on the values of equitable total chromatic number of factors G and H, equitable total chromatic number of corona G◦H is equal to Δ(G◦H)+1. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 4; 1147-1163
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Facial [r,s,t]-Colorings of Plane Graphs
Autorzy:
Czap, Július
Šugerek, Peter
Jendrol’, Stanislav
Valiska, Juraj
Powiązania:
https://bibliotekanauki.pl/articles/31343366.pdf
Data publikacji:
2019-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
plane graph
boundary walk
edge-coloring
vertex-coloring
total-coloring
Opis:
Let $G$ be a plane graph. Two edges are facially adjacent in $G$ if they are consecutive edges on the boundary walk of a face of $G$. Given nonnegative integers $r$, $s$, and $t$, a facial $[r, s, t]$-coloring of a plane graph $G = (V,E)$ is a mapping $f : V \cup E \rightarrow {1, . . ., k} $ such that $ |f(v_1) − f(v_2)| \ge r $ for every two adjacent vertices $ v_1 $ and $ v_2 $, $ | f(e_1) − f(e_2)| \ge s $ for every two facially adjacent edges $ e_1 $ and $ e_2 $, and $ | f(v) − f(e)| \ge t $ for all pairs of incident vertices $ v $ and edges $ e $. The facial $[r, s, t]$-chromatic number $ \overline{ \chi }_{r,s,t} (G) $ of $ G $ is defined to be the minimum $k$ such that $G$ admits a facial $[r, s, t]$-coloring with colors $1, . . ., k$. In this paper we show that $ \overline{ \chi }_{r,s,t} (G) \le 3r + 3s + t + 1 $ for every plane graph $G$. For some triplets $ [r, s, t] $ and for some families of plane graphs this bound is improved. Special attention is devoted to the cases when the parameters $r$, $s$, and $t$ are small.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 3; 629-645
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Fractional (\( \mathcal{P} , \mathcal{Q} \))-Total List Colorings of Graphs
Autorzy:
Kemnitz, Arnfried
Mihók, Peter
Voigt, Margit
Powiązania:
https://bibliotekanauki.pl/articles/30146708.pdf
Data publikacji:
2013-03-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
graph property
total coloring
(P,Q)-total coloring
fractional coloring
fractional (P,Q)-total chromatic number
circular coloring
circular (P,Q)-total chromatic number
list coloring
(P,Q)-total (a
b)-list colorings
Opis:
Let $ r, s \in \mathbb{N}$, $r \ge s$, and \( \mathcal{P} \) and \( \mathcal{Q} \) be two additive and hereditary graph properties. A \( (P,Q) \)-total $(r, s)$-coloring of a graph $G = (V,E)$ is a coloring of the vertices and edges of $G$ by $s$-element subsets of $ \mathbb{Z}_r$ such that for each color $i$, $0 \le i \le r − 1$, the vertices colored by subsets containing $i$ induce a subgraph of $G$ with property \( \mathcal{P} \), the edges colored by subsets containing $i$ induce a subgraph of $G$ with property \( \mathcal{Q} \), and color sets of incident vertices and edges are disjoint. The fractional \( (\mathcal{P}, \mathcal{Q})\)-total chromatic number $ \chi_{f,P,Q}^{''}(G)$ of $G$ is defined as the infimum of all ratios $r//s$ such that $G$ has a \( ( \mathcal{P}, \mathcal{Q})\)-total $(r, s)$-coloring. A \( ( \mathcal{P}, \mathcal{Q} \)-total independent set $ T = V_T \cup E_T \subseteq V \cup E$ is the union of a set $V_T$ of vertices and a set $E_T$ of edges of $G$ such that for the graphs induced by the sets $V_T$ and $E_T$ it holds that \( G[ V_T ] \in \mathcal{ P } \), \( G[ E_T ] \in \mathcal{Q} \), and $ G[ V_T ] $ and $ G[ E_T ] $ are disjoint. Let \( T_{ \mathcal{P} , \mathcal{Q} } \) be the set of all \( (\mathcal{P} ,\mathcal{Q})\)-total independent sets of $G$. Let $L(x)$ be a set of admissible colors for every element $ x \in V \cup E $. The graph $G$ is called \( (\mathcal{P} , \mathcal{Q}) \)-total $(a, b)$-list colorable if for each list assignment $L$ with $|L(x)| = a$ for all $x \in V \cup E$ it is possible to choose a subset $ C(x) \subseteq L(x)$ with $|C(x)| = b$ for all $ x \in V \cup E$ such that the set $ T_i $ which is defined by $ T_i = {x \in V \cup E : i \in C(x) } $ belongs to \( T_{ \mathcal{P},\mathcal{Q}}\) for every color $i$. The \( (\mathcal{P}, \mathcal{Q})\)- choice ratio \( \text{chr}_{\mathcal{P},\mathcal{Q}}(G)\) of $G$ is defined as the infimum of all ratios $a//b$ such that $G$ is \( (\mathcal{P},\mathcal{Q})\)-total $(a, b)$-list colorable. We give a direct proof of \( \chi_{ f,\mathcal{P},\mathcal{Q} }^{ \prime \prime } (G) = \text{chr}_{ \mathcal{P} ,\mathcal{Q} }(G)\) for all simple graphs $G$ and we present for some properties \( \mathcal{P} \) and \( \mathcal{Q} \) new bounds for the \( (\mathcal{P}, \mathcal{Q})\)-total chromatic number and for the \((\mathcal{P},\mathcal{Q})\)-choice ratio of a graph $G$.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 1; 167-179
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Generalized Fractional and Circular Total Colorings of Graphs
Autorzy:
Kemnitz, Arnfried
Marangio, Massimiliano
Mihók, Peter
Oravcová, Janka
Soták, Roman
Powiązania:
https://bibliotekanauki.pl/articles/31339338.pdf
Data publikacji:
2015-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
graph property
(P,Q)-total coloring
circular coloring
fractional coloring
fractional (P,Q)-total chromatic number
circular (P,Q)- total chromatic number
Opis:
Let \( \mathcal{P} \) and \( \mathcal{Q} \) be additive and hereditary graph properties, $ r, s \in \mathbb{N}$, $ r \ge s $, and $ [\mathbb{Z}_r]^s $ be the set of all s-element subsets of $\mathbb{Z}_r $. An ($r$, $s$)-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of $G$ is an assignment $ h : V (G) \cup E(G) \rightarrow [\mathbb{Z}_r]^s $ such that for each $ i \in \mathbb{Z}_r $ the following holds: the vertices of $G$ whose color sets contain color $i$ induce a subgraph of $G$ with property \( \mathcal{P} \), edges with color sets containing color $i$ induce a subgraph of $G$ with property \( \mathcal{Q} \), and the color sets of incident vertices and edges are disjoint. If each vertex and edge of $G$ is colored with a set of $s$ consecutive elements of $ \mathbb{Z}_r $ we obtain an ($r$, $s$)-circular (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of $G$. In this paper we present basic results on ($r$, $s$)-fractional/circular (\( \mathcal{P} \),\( \mathcal{Q} \))-total colorings. We introduce the fractional and circular (\( \mathcal{P} \),\( \mathcal{Q}\))-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 3; 517-532
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Generalized Fractional Total Colorings of Complete Graph
Autorzy:
Karafová, Gabriela
Powiązania:
https://bibliotekanauki.pl/articles/30145422.pdf
Data publikacji:
2013-09-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
fractional coloring
total coloring
complete graphs
Opis:
An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let $P$ and $Q$ be two additive and hereditary graph properties and let $r,s$ be integers such that $r\geq s$. Then an $\frac{r}{s}$-fractional $(P,Q)$-total coloring of a finite graph $G=(V,E)$ is a mapping $f$, which assigns an $s$-element subset of the set $\{1,2,...,r\}$ to each vertex and each edge, moreover, for any color $i$ all vertices of color $i$ induce a subgraph of property $P$, all edges of color $i$ induce a subgraph of property $Q$ and vertices and incident edges have assigned disjoint sets of colors. The minimum ratio $\frac{r}{s}$ of an $\frac{r}{s}$-fractional $(P,Q)$-total coloring of $G$ is called fractional $(P,Q)$-total chromatic number $\chi_{f,P,Q}^{''}(G)=\frac{r}{s}$. Let $k=$ sup$\{i:K_{i+1}\in P\}$ and $l=$ sup$\{i:K_{i+1}\in Q\}$. We show for a complete graph $K_{n}$ that if $l\geq k+2$ then $\chi_{f,P,Q}^{''}(K_{n})=\frac{n}{k+1}$ for a sufficiently large $n$.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 4; 665-676
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Generalized Fractional Total Colorings of Graphs
Autorzy:
Karafová, Gabriela
Soták, Roman
Powiązania:
https://bibliotekanauki.pl/articles/31339383.pdf
Data publikacji:
2015-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
fractional coloring
total coloring
automorphism group
Opis:
Let \( \mathcal{P} \) and \( \mathcal{Q} \) be additive and hereditary graph properties and let $r$, $s$ be integers such that $ r \ge s $. Then an $ r/s$-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of a finite graph $ G = (V, E) $ is a mapping $f$, which assigns an $s$-element subset of the set $ {1, 2, . . ., r}$ to each vertex and each edge, moreover, for any color $i$ all vertices of color $i$ induce a subgraph with property \( \mathcal{P} \), all edges of color $i$ induce a subgraph with property \( \mathcal{Q} \) and vertices and incident edges have been assigned disjoint sets of colors. The minimum ratio of an \( \frac{r}{s} \)-fractional (\( \mathcal{P} \),\( \mathcal{Q} \))-total coloring of G is called fractional (\( \mathcal{P} \), \( \mathcal{Q} \))-total chromatic number \( \chi_{f, \mathcal{P} ,\mathcal{Q} }^{ \prime \prime } (G) = \frac{r}{s} \). We show in this paper that \( \chi_{f, \mathcal{P} ,\mathcal{Q} }^{ \prime \prime } \) of a graph \( G \) with \( o(V (G)) \) vertex orbits and \( o(E(G)) \) edge orbits can be found as a solution of a linear program with integer coefficients which consists only of \( o(V (G)) + o(E(G)) \) inequalities.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 3; 463-473
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Intersection graph of gamma sets in the total graph
Autorzy:
Chelvam, T.
Asir, T.
Powiązania:
https://bibliotekanauki.pl/articles/743214.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total graph
gamma sets
intersection graph
Hamiltonian
coloring
connectivity
domination number
Opis:
In this paper, we consider the intersection graph $I_{Γ}(ℤₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ}(ℤₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ}(ℤₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ}(ℤₙ)$.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 2; 341-356
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Neighbor Product Distinguishing Total Colorings of Planar Graphs with Maximum Degree at least Ten
Autorzy:
Dong, Aijun
Li, Tong
Powiązania:
https://bibliotekanauki.pl/articles/32227944.pdf
Data publikacji:
2021-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total coloring
neighbor product distinguishing coloring
planar graph
Opis:
A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, . . ., k}. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) ≠ p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By X″∏(G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that Δ(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with Δ(G) ≥ 10. Moreover, for planar graph G with Δ(G) ≥ 11, it is neighbor product distinguishing (Δ(G) + 2)-total colorable, and the upper bound Δ(G) + 2 is tight.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 4; 981-999
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles
Autorzy:
Zhao, Xue
Xu, Chang-Qing
Powiązania:
https://bibliotekanauki.pl/articles/32083807.pdf
Data publikacji:
2020-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
neighbor sum distinguishing total coloring
discharging method
planar graph
Opis:
For a given graph $ G = (V (G), E(G)) $, a proper total coloring $ \phi : V (G) \cup E(G) $ $ \rightarrow {1, 2, . . ., k} $ is neighbor sum distinguishing if $ f(u) \ne f(v) $ for each edge $ uv \in E(G) $, where $ f(v) = \Sigma_{ uv \in E(G) } $ $ \phi (uv) + \phi (v) $, $ v \in V (G) $. The smallest integer $k$ in such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $ \chi_\Sigma^{''} (G) $. Pilśniak and Woźniak first introduced this coloring and conjectured that $ \chi_\Sigma^{''}(G) \le \Delta (G)+3 $ for any graph with maximum degree $ \Delta (G) $. In this paper, by using the discharging method, we prove that for any planar graph $G$ without 5-cycles, $ \chi_\Sigma^{''} (G) \le \text{max} \{ \Delta (G)+2, 10 \} $. The bound $ \Delta (G) + 2 $ is sharp. Furthermore, we get the exact value of $ \chi_\Sigma^{''} (G) $ if $ \Delta (G) \ge 9 $.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 1; 243-253
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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