Packing chromatic number, $$\mathbf (1, 1, 2, 2) $$(1,1,2,2)-colorings, and characterizing the Petersen graph
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DOI: 10.1007/s00010-016-0461-8
Abstract: The packing chromatic number $$\chi _{\rho }(G)$$χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets $$\Pi _1,\ldots ,\Pi _k$$Π1,…,Πk, where $$\Pi _i$$Πi,… read more here.
Keywords: graph; number; petersen graph; packing chromatic ... See more keywords
On the packing coloring of base-3 Sierpiński graphs and H-graphs
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DOI: 10.1007/s00010-020-00747-w
Abstract: For a nondecreasing sequence of integers $$S=(s_1, s_2, \ldots )$$ S = ( s 1 , s 2 , … ) an S -packing k -coloring of a graph G is a mapping from V… read more here.
Keywords: graphs; base sierpi; packing coloring; sierpi ski ... See more keywords
Radio k-chromatic Number of Full m-ary Trees
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DOI: 10.1007/s00224-021-10056-7
Abstract: For a simple connected graph G = (V (G),E(G)) and a positive integer k, a radio k-labelling of G is a mapping $f \colon V(G)\rightarrow \{0,1,2,\ldots \}$ such that $|f(u)-f(v)|\geqslant k+1-d(u,v)$ for each pair of… read more here.
Keywords: radio chromatic; ary trees; chromatic number; number full ... See more keywords
Group Colorings and DP-Colorings of Multigraphs Using Edge-Disjoint Decompositions
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DOI: 10.1007/s00373-021-02345-2
Abstract: In (J Graph Theory 4:241–242, 1980), Burr proved that $$\chi (G)\le m_1m_2 \ldots m_k$$ if and only if G is the edge-disjoint union of k graphs $$G_1,G_2,\ldots ,G_k$$ such that $$\chi (G_i)\le m_i$$ for $$1\le… read more here.
Keywords: number; chromatic number; group; edge disjoint ... See more keywords
The fractional chromatic number of the plane
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DOI: 10.1007/s00493-016-3380-3
Abstract: The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance… read more here.
Keywords: fractional chromatic; number; chromatic number; number plane ... See more keywords
Infinitely connected subgraphs in graphs of uncountable chromatic number
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DOI: 10.1007/s00493-016-3436-4
Abstract: Erdős and Hajnal conjectured in 1966 that every graph of uncountable chromatic number contains a subgraph of infinite connectivity. We prove that every graph of uncountable chromatic number has a subgraph which has uncountable chromatic… read more here.
Keywords: connected subgraphs; uncountable chromatic; subgraphs graphs; infinitely connected ... See more keywords
Chromatic Number of Ordered Graphs with Forbidden Ordered Subgraphs
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DOI: 10.1007/s00493-017-3593-0
Abstract: It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a… read more here.
Keywords: left right; graphs; number ordered; ordered graphs ... See more keywords
A note on orientation and chromatic number of graphs
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DOI: 10.1007/s10878-016-0094-9
Abstract: Let D be any edge orientation of a graph G. We denote by $$\Delta _k(D)$$Δk(D) the maximum value t for which there exists a directed path $$v_1, \ldots , v_k$$v1,…,vk such that $$d^{out}(v_k)=t$$dout(vk)=t, where $$d^{out}(v_k)$$dout(vk)… read more here.
Keywords: orientation; orientation chromatic; number graphs; note orientation ... See more keywords
Chromatic Number with Several Forbidden Distances in the Space with the ℓq-Metric
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DOI: 10.1007/s10958-017-3592-0
Abstract: We study the chromatic number χ¯Xρk$$ \overline{\chi}\left(X;\rho; k\right) $$ of a metric space X with a metric ρ and k forbidden distances. We obtain an estimate of the form χ¯ℝnρk≥BkCn$$ \overline{\chi}\left({\mathbb{R}}^n;\rho; k\right)\ge {(Bk)}^{Cn} $$ for… read more here.
Keywords: forbidden distances; space metric; chromatic number; number several ... See more keywords
On the Connection Between the Chromatic Number of a Graph and the Number of Cycles Covering a Vertex or an Edge
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DOI: 10.1007/s10958-018-3853-6
Abstract: We prove several tight bounds on the chromatic number of a graph in terms of the minimum number of simple cycles covering a vertex or an edge of this graph. Namely, we prove that X(G) ≤ k… read more here.
Keywords: edge; graph; number; cycles covering ... See more keywords
The Distinguishing Number and Distinguishing Chromatic Number for Posets
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DOI: 10.1007/s11083-021-09583-2
Abstract: In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice $L$ and its set $Q_L$ of join-irreducibles, we use classic lattice theory… read more here.
Keywords: distinguishing number; distinguishing chromatic; number posets; number ... See more keywords