Přemysl Holub

The rainbow connection number of 2-connected graphs

Abstract The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al., we prove that the rainbow connection number of every 2-connected graph with n vertices is at most ⌈ n / 2 ⌉ . The bound is optimal.

Packing chromatic number of distance graphs

The packing chromatic number ? ? ( G ) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X 1 , ? , X k where vertices in X i have pairwise distance greater than i . We study the packing chromatic number of infinite distance graphs G ( Z , D ) , i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i , j ? Z are adjacent if and only if | i - j | ? D .In this paper we focus on distance graphs with D = { 1 , t } . We improve some results of Togni who initiated the study. It is shown that ? ? ( G ( Z , D ) ) ? 35 for sufficiently large odd t and ? ? ( G ( Z , D ) ) ? 56 for sufficiently large even t . We also give a lower bound 12 for t ? 9 and tighten several gaps for ? ? ( G ( Z , D ) ) with small t .

Rainbow connection and forbidden subgraphs

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc ( G ) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant k F such that, for every connected F -free graph G , rc ( G ) ? diam ( G ) + k F , where diam ( G ) is the diameter of G . In this paper, we give a complete answer for | F | ? { 1 , 2 } .

Degree diameter problem on honeycomb networks

The degree diameter problem involves finding the largest graph (in terms of the number of vertices) subject to constraints on the degree and the diameter of the graph. Beyond the degree constraint there is no restriction on the number of edges (apart from keeping the graph simple) so the resulting graph may be thought of as being embedded in the complete graph. In a generalization of this problem, the graph is considered to be embedded in some connected host graph, in this paper the honeycomb network. We consider embedding the graph in the k -dimensional honeycomb grid and provide upper and lower bounds for the optimal graph. The particular cases of dimensions 2 and 3 are examined in detail.

Radio labelings of distance graphs

Motivated by the Channel Assignment Problem, we study radio k-labelings of graphs. A radio k-labeling of a connected graph G is an assignment c of non-negative integers to the vertices of G such that |c(x)-c(y)|>=k+1-d(x,y), for any two vertices x and y, x y, where d(x,y) is the distance between x and y in G. In this paper, we study radio k-labelings of distance graphs, i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i,j@?Z are adjacent if and only if |i-j|@?D. We give some lower and upper bounds for radio k-labelings of distance graphs with distance sets D(1,2,...,t), D(1,t) and D(t-1,t) for any positive integer t>1.

On forbidden subgraphs and rainbow connection in graphs with minimum degree 2

Abstract A connected edge-colored graph G is said to be rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors, and the rainbow connection number rc ( G ) of G is the minimum number of colors that can make G rainbow-connected. We consider families F of connected graphs for which there is a constant k F such that every connected F -free graph G with minimum degree at least 2 satisfies rc ( G ) ≤ diam ( G ) + k F , where diam ( G ) is the diameter of G . In this paper, we give a complete answer for | F | = 1 , and a partial answer for | F | = 2 .

Finite families of forbidden subgraphs for rainbow connection in graphs

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc ( G ) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant k F such that, for every connected F -free graph G , rc ( G ) ź diam ( G ) + k F , where diam ( G ) is the diameter of G . In the paper, we finalize our previous considerations and give a complete answer for any finite family F .

3-Coloring Triangle-Free Planar Graphs with a Precolored 9-Cycle

Given a triangle-free planar graph G and a cycle C of length 9 in G, we characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results for the length of C up to 8.

Characterizing forbidden pairs for rainbow connection in graphs with minimum degree 2

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc ( G ) of G is the minimum number of colors that are needed in order to make G rainbow connected. In this paper, we complete the discussion of pairs ( X , Y ) of connected graphs for which there is a constant k X Y such that, for every connected ( X , Y ) -free graph G with minimum degree at least 2, rc ( G ) ? diam ( G ) + k X Y (where diam ( G ) is the diameter of G ), by giving a complete characterization. In particular, we show that for every connected ( Z 3 , S 3 , 3 , 3 ) -free graph G with ? ( G ) ? 2 , rc ( G ) ? diam ( G ) + 156 , and, for every connected ( S 2 , 2 , 2 , N 2 , 2 , 2 ) -free graph G with ? ( G ) ? 2 , rc ( G ) ? diam ( G ) + 72 .

4-colorability of P 6 -free graphs

Abstract In this paper we will study the complexity of 4-colorability in subclasses of P 6 -free graphs. The well known k-colorability problem is NP-complete. It has been shown that if k-colorability is solvable in polynomial time for an induced H-free graph, then every component of H is a path. Recently, Huang [S. Huang, Improved Complexity Results on k-Coloring P t -Free Graphs, Proc. MFCS, LNCS 8087 (2013) 551558] has shown several improved complexity results on k-coloring P t -free graphs, where P t is an induced path on t vertices. In summer 2014 only the case k = 4 , t = 6 remained open for all k ≥ 4 and all t ≥ 6 . Huang conjectures that 4-colorability of P 6 -free graphs can be decided in polynomial time. This conjecture has shown to be true for the class of ( P 6 , banner)-free graphs by Huang [S. Huang, Improved Complexity Results on k-Coloring P t -Free Graphs, Proc. MFCS, LNCS 8087 (2013) 551558] and for the class of ( P 6 , C 5 )-free graphs by Chudnovsky et al. [M. Chudnovsky, P. Maceli, J. Stacho, and M. Zhong, 4-coloring P 6 -free graphs with no induced 5-cycles, submitted]. In this paper we show that the conjecture also holds for the class of ( P 6 , bull, Z 2 )-free graphs, for the class of ( P 6 , bull, kite)-free graphs, and for the class of ( P 6 , chair)-free graphs.