Interpretable random forest model for identification of edge 3-uncolorable cubic graphs

Adam Dudáš; Bianka Modrovičová

Displaying similar documents to “Interpretable random forest model for identification of edge 3-uncolorable cubic graphs”

Zero-one laws for graphs with edge probabilities decaying with distance. Part II

Saharon Shelah (2005)

Fundamenta Mathematicae

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Let Gₙ be the random graph on [n] = 1,...,n with the probability of i,j being an edge decaying as a power of the distance, specifically the probability being $p_{| i - j |} = 1 / {| i - j |}^{α}$ , where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a “finite compactness”. Lastly, as a consequence, we prove that the zero-one law (for first...

Edge maximal $C_{2 k + 1}$ -edge disjoint free graphs

M.S.A. Bataineh, M.M.M. Jaradat (2012)

Discussiones Mathematicae Graph Theory

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For two positive integers r and s, 𝓖(n;r,s) denotes to the class of graphs on n vertices containing no r of s-edge disjoint cycles and f(n;r,s) = max{𝓔(G):G ∈ 𝓖(n;r,s)}. In this paper, for integers r ≥ 2 and k ≥ 1, we determine f(n;r,2k+1) and characterize the edge maximal members in 𝓖(n;r,2k+1).

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let $G$ be an undirected simple connected graph, and $e = u v$ be an edge of $G$ . Let $N_{G} (e)$ be the subgraph of $G$ induced by the set of all vertices of $G$ which are not incident to $e$ but are adjacent to $u$ or $v$ . Let $𝒩_{e}$ be the class of all graphs $H$ such that, for some graph $G$ , $N_{G} (e) ≅ H$ for every edge $e$ of $G$ . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in $𝒩_{e}$ . Balasubramanian and Alsardary [1] obtained some other graphs in $𝒩_{e}$ . In this paper we given some new graphs in $𝒩_{e}$ .

Generalized edge-chromatic numbers and additive hereditary properties of graphs

Michael J. Dorfling, Samantha Dorfling (2002)

Discussiones Mathematicae Graph Theory

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let and be hereditary properties of graphs. The generalized edge-chromatic number $ρ_{}^{'} ()$ is defined as the least integer n such that ⊆ n. We investigate the generalized edge-chromatic numbers of the properties → H, ₖ, ₖ, *ₖ, ₖ and ₖ.

On generating snarks

Busiso P. Chisala (1998)

Discussiones Mathematicae Graph Theory

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We discuss the construction of snarks (that is, cyclically 4-edge connected cubic graphs of girth at least five which are not 3-edge colourable) by using what we call colourable snark units and a welding process.

1-planar graphs with girth at least 6 are (1,1,1,1)-colorable

Lili Song, Lei Sun (2023)

Czechoslovak Mathematical Journal

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A graph is 1-planar if it can be drawn in the Euclidean plane so that each edge is crossed by at most one other edge. A 1-planar graph on $n$ vertices is optimal if it has $4 n - 8$ edges. We prove that 1-planar graphs with girth at least 6 are (1,1,1,1)-colorable (in the sense that each of the four color classes induces a subgraph of maximum degree one). Inspired by the decomposition of 1-planar graphs, we conjecture that every 1-planar graph is (2,2,2,0,0)-colorable.

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

Yury Metelsky, Kseniya Schemeleva, Frank Werner (2017)

Discussiones Mathematicae Graph Theory

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We characterize the class [...] L32 $L_{3}^{2}$ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_{3}^{2}$ in the class of threshold graphs, where n is the number of vertices of a tested graph.

Zero-one laws for graphs with edge probabilities decaying with distance. Part I

Saharon Shelah (2002)

Fundamenta Mathematicae

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Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability $p_{| i - j |} = 1 / {| i - j |}^{α}$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..

Maximum Edge-Colorings Of Graphs

Stanislav Jendrol’, Michaela Vrbjarová (2016)

Discussiones Mathematicae Graph Theory

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An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree dG(v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index [...] χr′(G) $χ_{r}^{'} (G)$ is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that [...] χr′(G)≤3 $χ_{r}^{'} (G) \leq 3$ for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with [...] χ1′(G)=i...

Destroying symmetry by orienting edges: complete graphs and complete bigraphs

Frank Harary, Michael S. Jacobson (2001)

Discussiones Mathematicae Graph Theory

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Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group. We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs $K_{s, t}$ , s ≤ t, this number does not always exist. We determine for s ≤ 4 the values of t for which this number does exist.

Clopen graphs

Stefan Geschke (2013)

Fundamenta Mathematicae

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A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their...

Upper bounds on the b-chromatic number and results for restricted graph classes

Mais Alkhateeb, Anja Kohl (2011)

Discussiones Mathematicae Graph Theory

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A b-coloring of a graph G by k colors is a proper vertex coloring such that every color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number $χ_{b} (G)$ is the maximum integer k for which G has a b-coloring by k colors. Moreover, the graph G is called b-continuous if G admits a b-coloring by k colors for all k satisfying $χ (G) \leq k \leq χ_{b} (G)$ . In this paper, we establish four general upper bounds on $χ_{b} (G)$ . We present results on the b-chromatic...

On Ramsey $(K_{1, 2}, C ₄)$ -minimal graphs

Tomás Vetrík, Lyra Yulianti, Edy Tri Baskoro (2010)

Discussiones Mathematicae Graph Theory

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For graphs F, G and H, we write F → (G,H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H. The graph F is Ramsey (G,H)-minimal if F → (G,H) but F* ↛ (G,H) for any proper subgraph F* ⊂ F. We present an infinite family of Ramsey $(K_{1, 2}, C ₄)$ -minimal graphs of any diameter ≥ 4.

Balanced problems on graphs with categorization of edges

Štefan Berežný, Vladimír Lacko (2003)

Discussiones Mathematicae Graph Theory

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Suppose a graph G = (V,E) with edge weights w(e) and edges partitioned into disjoint categories S₁,...,Sₚ is given. We consider optimization problems on G defined by a family of feasible sets (G) and the following objective function: $L ₅ (D) = m a x_{1 \leq i \leq p} (m a x_{e \in S_{i} \cap D} w (e) - m i n_{e \in S_{i} \cap D} w (e))$ For an arbitrary number of categories we show that the L₅-perfect matching, L₅-a-b path, L₅-spanning tree problems and L₅-Hamilton cycle (on a Halin graph) problem are NP-complete. We also summarize polynomiality results concerning above objective functions...

Radio numbers for generalized prism graphs

Paul Martinez, Juan Ortiz, Maggy Tomova, Cindy Wyels (2011)

Discussiones Mathematicae Graph Theory

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A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted $Z_{n, s}$ , s ≥ 1, n ≥ s, have vertex set (i,j) | i = 1,2 and j = 1,...,n and edge set ((i,j),(i,j ±1)) ∪ ((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋. In this paper we determine...

A Littlewood-Paley-Stein estimate on graphs and groups

Nick Dungey (2008)

Studia Mathematica

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We establish the boundedness in $L^{q}$ spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.

Bounding the Openk-Monopoly Number of Strong Product Graphs

Dorota Kuziak, Iztok Peterin, Ismael G. Yero (2018)

Discussiones Mathematicae Graph Theory

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Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ, and let k ∈ 1 − ⌈δ/2⌉, . . . , ⌊δ/2⌋ be an integer. Given a set M ⊂ V, a vertex v of G is said to be k-controlled by M if [...] δM(v)≥δG(v)2+k $δ_{M} (v) \geq \frac{δ_{G} (v)}{2} + k$ , where δM(v) represents the number of neighbors of v in M and δG(v) the degree of v in G. A set M is called an open k-monopoly if every vertex v of G is k-controlled by M. The minimum cardinality of any open k-monopoly is the open k-monopoly number of G. In this...

The Chromatic Number of Random Intersection Graphs

Katarzyna Rybarczyk (2017)

Discussiones Mathematicae Graph Theory

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We study problems related to the chromatic number of a random intersection graph G (n,m, p). We introduce two new algorithms which colour G (n,m, p) with almost optimum number of colours with probability tending to 1 as n → ∞. Moreover we find a range of parameters for which the chromatic number of G (n,m, p) asymptotically equals its clique number.

Fall coloring of graphs I

Rangaswami Balakrishnan, T. Kavaskar (2010)

Discussiones Mathematicae Graph Theory

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A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number $χ_{f} (G)$ of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, $χ (G) \leq χ_{f} (G)$ . In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and $χ_{f} (G)$ . We also obtain the smallest non-fall...