Classes of hypergraphs with sum number one

Hanns-Martin Teichert

Displaying similar documents to “Classes of hypergraphs with sum number one”

The sum number of d-partite complete hypergraphs

Hanns-Martin Teichert (1999)

Discussiones Mathematicae Graph Theory

Similarity:

A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph $⁺_{d} (S) = (V,)$ , where V = S and $= v ₁, . . ., v_{d} : (i \neq j \Rightarrow v_{i} \neq v_{j}) \land \sum_{i = 1}^{d} v_{i} \in S$ . For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $w ₁, . . ., w_{σ} \notin V$ such that $\cup w ₁, . . ., w_{σ}$ is a sum hypergraph. In this paper, we prove $σ (_{n ₁, . . ., n_{d}}^{d}) = 1 + \sum_{i = 1}^{d} (n_{i} - 1) + m i n 0, ⌈ 1 / 2 (\sum_{i = 1}^{d - 1} (n_{i} - 1) - n_{d}) ⌉$ , where $_{n ₁, . . ., n_{d}}^{d}$ denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

Neighbor sum distinguishing list total coloring of IC-planar graphs without 5-cycles

Donghan Zhang (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $G = (V (G), E (G))$ be a simple graph and $E_{G} (v)$ denote the set of edges incident with a vertex $v$ . A neighbor sum distinguishing (NSD) total coloring $φ$ of $G$ is a proper total coloring of $G$ such that $\sum_{z \in E_{G} (u) \cup {u}} φ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} φ (z)$ for each edge $u v \in E (G)$ . Pilśniak and Woźniak asserted in 2015 that each graph with maximum degree $Δ$ admits an NSD total $(Δ + 3)$ -coloring. We prove that the list version of this conjecture holds for any IC-planar graph with $Δ \geq 11$ but without $5$ -cycles by applying the Combinatorial Nullstellensatz.

Generalized list colourings of graphs

Mieczysław Borowiecki, Ewa Drgas-Burchardt, Peter Mihók (1995)

Discussiones Mathematicae Graph Theory

Similarity:

We prove: (1) that $c h_{P} (G) - χ_{P} (G)$ can be arbitrarily large, where $c h_{P} (G)$ and $χ_{P} (G)$ are P-choice and P-chromatic numbers, respectively, (2) the (P,L)-colouring version of Brooks’ and Gallai’s theorems.

On distinguishing and distinguishing chromatic numbers of hypercubes

Werner Klöckl (2008)

Discussiones Mathematicae Graph Theory

Similarity:

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number $χ_{D} (G)$ of G. Extending these concepts to infinite graphs we prove that $D (Q_{ℵ} ₀) = 2$ and $χ_{D} (Q_{ℵ} ₀) = 3$ , where $Q_{ℵ} ₀$ denotes the hypercube of countable dimension. We also show that $χ_{D} (Q ₄) = 4$ , thereby completing the investigation of finite hypercubes with respect to $χ_{D}$ . Our...

Note on improper coloring of $1$ -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

Similarity:

A graph $G = (V, E)$ is called improperly $(d_{1}, \dots, d_{k})$ -colorable if the vertex set $V$ can be partitioned into subsets $V_{1}, \dots, V_{k}$ such that the graph $G [V_{i}]$ induced by the vertices of $V_{i}$ has maximum degree at most $d_{i}$ for all $1 \leq i \leq k$ . In this paper, we mainly study the improper coloring of $1$ -planar graphs and show that $1$ -planar graphs with girth at least $7$ are $(2, 0, 0, 0)$ -colorable.

On subgraphs without large components

Glenn G. Chappell, John Gimbel (2017)

Mathematica Bohemica

Similarity:

We consider, for a positive integer $k$ , induced subgraphs in which each component has order at most $k$ . Such a subgraph is said to be $k$ -divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a $k$ -divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for...

Edge-colouring of graphs and hereditary graph properties

Samantha Dorfling, Tomáš Vetrík (2016)

Czechoslovak Mathematical Journal

Similarity:

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph $G$ , a hereditary graph property $𝒫$ and $l \geq 1$ we define $χ_{𝒫, l}^{'} (G)$ to be the minimum number of colours needed to properly colour the edges of $G$ , such that any subgraph of $G$ induced by edges coloured by (at most) $l$ colours is in $𝒫$ . We present a necessary and sufficient condition for the existence of $χ_{𝒫, l}^{'} (G)$ . We focus on edge-colourings of graphs with respect to the hereditary...

Sum labellings of cycle hypergraphs

Hanns-Martin Teichert (2000)

Discussiones Mathematicae Graph Theory

Similarity:

A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph $_{d ̲, [d ̅]} (S) = (V,)$ where V = S and $= e \subseteq S : d ̲ \leq | e | \leq [d ̅] \land \sum_{v \in e} v \in S$ . For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices $y ₁, . . ., y_{σ} \notin V$ such that $\cup y ₁, . . ., y_{σ}$ is a sum hypergraph. Generalizing the graph Cₙ we obtain d-uniform hypergraphs where any d consecutive vertices of Cₙ form an edge. We determine sum numbers and investigate properties of sum labellings...

Acyclic 4-choosability of planar graphs without 4-cycles

Yingcai Sun, Min Chen (2020)

Czechoslovak Mathematical Journal

Similarity:

A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle in $G$ . In other words, each cycle of $G$ must be colored with at least three colors. Given a list assignment $L = {L (v) : v \in V}$ , if there exists an acyclic coloring $π$ of $G$ such that $π (v) \in L (v)$ for all $v \in V$ , then we say that $G$ is acyclically $L$ -colorable. If $G$ is acyclically $L$ -colorable for any list assignment $L$ with $| L (v) | \geq k$ for all $v \in V$ , then $G$ is acyclically $k$ -choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without...

On $g_{c}$ -colorings of nearly bipartite graphs

Yuzhuo Zhang, Xia Zhang (2018)

Czechoslovak Mathematical Journal

Similarity:

Let $G$ be a simple graph, let $d (v)$ denote the degree of a vertex $v$ and let $g$ be a nonnegative integer function on $V (G)$ with $0 \leq g (v) \leq d (v)$ for each vertex $v \in V (G)$ . A $g_{c}$ -coloring of $G$ is an edge coloring such that for each vertex $v \in V (G)$ and each color $c$ , there are at least $g (v)$ edges colored $c$ incident with $v$ . The $g_{c}$ -chromatic index of $G$ , denoted by $χ_{g_{c}}^{'} (G)$ , is the maximum number of colors such that a $g_{c}$ -coloring of $G$ exists. Any simple graph $G$ has the $g_{c}$ -chromatic index equal to $δ_{g} (G)$ or $δ_{g} (G) - 1$ , where $δ_{g} (G) = {min}_{v \in V (G)} ⌊ d (v) / g (v) ⌋$ . A graph $G$ is nearly bipartite,...

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

Similarity:

In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Proper connection number of bipartite graphs

Jun Yue, Meiqin Wei, Yan Zhao (2018)

Czechoslovak Mathematical Journal

Similarity:

An edge-colored graph $G$ is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph $G$ , denoted by $pc (G)$ , is the smallest number of colors that are needed to color the edges of $G$ in order to make it proper connected. In this paper, we obtain the sharp upper bound for $pc (G)$ of a general bipartite graph $G$ and a series of extremal graphs. Additionally, we give a proper $2$ -coloring for a connected bipartite graph $G$ having $δ (G) \geq 2$ and a dominating...

Persistency in the Traveling Salesman Problem on Halin graphs

Vladimír Lacko (2000)

Discussiones Mathematicae Graph Theory

Similarity:

For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition $E_{A l l}$ , $E_{S o m e}$ , $E_{N o n e}$ of the edge set E, where: $E_{A l l}$ = e ∈ E, e belongs to all optimum solutions, $E_{N o n e}$ = e ∈ E, e does not belong to any optimum solution and $E_{S o m e}$ = e ∈ E, e belongs to some but not to all optimum solutions.

Generalized connectivity of some total graphs

Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)

Czechoslovak Mathematical Journal

Similarity:

We study the generalized $k$ -connectivity $κ_{k} (G)$ as introduced by Hager in 1985, as well as the more recently introduced generalized $k$ -edge-connectivity $λ_{k} (G)$ . We determine the exact value of $κ_{k} (G)$ and $λ_{k} (G)$ for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case $k = 3$ .