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K3-Worm Colorings of Graphs: Lower Chromatic Number and Gaps in the Chromatic Spectrum

100%
Bujtás C., Tuza Z.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 3
759-772
EN
A K3-WORM coloring of a graph G is an assignment of colors to the vertices in such a way that the vertices of each K3-subgraph of G get precisely two colors. We study graphs G which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer. 219 (2014) 161-173] by proving that for every integer k ≥ 3 there exists a K3-WORM-colorable graph in which the minimum number of colors is exactly k. There also exist K3-WORM colorable graphs which have a K3-WORM coloring with two colors and also with k colors but no coloring with any of 3, . . . , k − 1 colors. We also prove that it is NP-hard to determine the minimum number of colors, and NP-complete to decide k-colorability for every k ≥ 2 (and remains intractable even for graphs of maximum degree 9 if k = 3). On the other hand, we prove positive results for d-degenerate graphs with small d, also including planar graphs.
2
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Domination Game Critical Graphs

81%
Bujtás C., Klavžar S., Košmrlj G.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 4
781-796
EN
The domination game is played on a graph G by two players who alternately take turns by choosing a vertex such that in each turn at least one previously undominated vertex is dominated. The game is over when each vertex becomes dominated. One of the players, namely Dominator, wants to finish the game as soon as possible, while the other one wants to delay the end. The number of turns when Dominator starts the game on G and both players play optimally is the graph invariant γg(G), named the game domination number. Here we study the γg-critical graphs which are critical with respect to vertex predomination. Besides proving some general properties, we characterize γg-critical graphs with γg = 2 and with γg = 3, moreover for each n we identify the (infinite) class of all γg-critical ones among the nth powers CnN of cycles. Along the way we determine γg(CnN) for all n and N. Results of a computer search for γg-critical trees are presented and several problems and research directions are also listed.
3
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Color-bounded hypergraphs, V: host graphs and subdivisions

81%
Bujtás C., Tuza Z., Voloshin V.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
|
nr 2
223-238
EN
A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set 𝓔 = {E₁,...,Eₘ}, together with integers $s_i$ and $t_i$ satisfying $1 ≤ s_i ≤ t_i ≤ |E_i|$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_i$ satisfies $s_i ≤ |φ(E_i)| ≤ t_i$. The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a "host graph", that means a graph G on the same vertex set X as ℋ, such that each $E_i$ induces a connected subgraph in G. In the current setting we fix a graph or multigraph G₀, and assume that the host graph G is obtained by some sequence of edge subdivisions, starting from G₀. The colorability problem is known to be NP-complete in general, and also when restricted to 3-uniform "mixed hypergraphs", i.e., color-bounded hypergraphs in which $|E_i| = 3$ and $1 ≤ s_i ≤ 2 ≤ t_i ≤ 3$ holds for all i ≤ m. We prove that for every fixed graph G₀ and natural number r, colorability is decidable in polynomial time over the class of r-uniform hypergraphs (and more generally of hypergraphs with $|E_i| ≤ r$ for all 1 ≤ i ≤ m) having a host graph G obtained from G₀ by edge subdivisions. Stronger bounds are derived for hypergraphs for which G₀ is a tree.
4
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3-consecutive c-colorings of graphs

71%
Bujtás C., Sampathkumar E., Tuza Z., Subramanya M., Dominic C.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
|
nr 3
393-405
EN
A 3-consecutive C-coloring of a graph G = (V,E) is a mapping φ:V → ℕ such that every path on three vertices has at most two colors. We prove general estimates on the maximum number $(χ̅)_{3CC}(G)$ of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with $(χ̅)_{3CC}(G) ≥ k$ for k = 3 and k = 4.
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