Ponències/Comunicacions de congressos
http://hdl.handle.net/2117/186068
20210516T02:49:44Z

On list kcoloring convex bipartite graphs
http://hdl.handle.net/2117/341144
On list kcoloring convex bipartite graphs
Díaz Cort, Josep; Yasar Diner, Oznur; Serna Iglesias, María José; Serra Albó, Oriol
List k–Coloring (LI kCOL) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..., k}. The problem is known to be NPhard even for k = 3 within the class of 3–regular planar bipartite graphs and for k = 4 within the class of chordal bipartite graphs. In 2015 Huang, Johnson and Paulusma asked for the complexity of LI 3COL in the class of chordal bipartite graphs. In this paper, we give a partial answer to this question by showing that LI kCOL is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the LI kCOL in the class of convex bipartite graphs. Finally, we show how our algorithm can be modified to solve the more general LI HCOL problem on convex bipartite graphs.
20210308T14:43:47Z
Díaz Cort, Josep
Yasar Diner, Oznur
Serna Iglesias, María José
Serra Albó, Oriol
List k–Coloring (LI kCOL) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..., k}. The problem is known to be NPhard even for k = 3 within the class of 3–regular planar bipartite graphs and for k = 4 within the class of chordal bipartite graphs. In 2015 Huang, Johnson and Paulusma asked for the complexity of LI 3COL in the class of chordal bipartite graphs. In this paper, we give a partial answer to this question by showing that LI kCOL is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the LI kCOL in the class of convex bipartite graphs. Finally, we show how our algorithm can be modified to solve the more general LI HCOL problem on convex bipartite graphs.

How to determine if a random graph with a fixed degree sequence has a giant component
http://hdl.handle.net/2117/186067
How to determine if a random graph with a fixed degree sequence has a giant component
Joos, Felix; Perarnau Llobet, Guillem; Rautenbach, Dieter; Reed, Bruce
The traditional ErdosRenyi model of a random network is of little use in modelling the type of complex networks which modern researchers study. In this graph, every pair of vertices is equally likely to be connected by an edge. However, 21st century networks are of diverse nature and usually exhibit inhomogeneity among their nodes. This motivates the study, for a fixed degree sequence D=(d1, ..., dn), of a uniformly chosen simple graph G(D) on {1, ..., n} where the vertex i has degree di. In this paper, we study the existence of a giant component in G(D). A heuristic argument suggests that a giant component in G(D) will exist provided that the sum of the squares of the degrees is larger than twice the sum of the degrees. In 1995, Molloy and Reed essentially proved this to be the case when the degree sequence D under consideration satisfies certain technical conditions [Random Structures & Algorithms, 6:161180]. This work has attracted considerable attention, has been extended to degree sequences under weaker conditions and has been applied to random models of a wide range of complex networks such as the World Wide Web or biological systems operating at a submolecular level. Nevertheless, the technical conditions on D restrict the applicability of the result to sequences where the vertices of high degree play no important role. This is a major problem since it is observed in many realworld networks, such as scalefree networks, that vertices of high degree (the socalled hubs) are present and play a crucial role. In this paper we characterize when a uniformly random graph with a fixed degree sequence has a giant component. Our main result holds for every degree sequence of length n provided that a minor technical condition is satisfied. The typical structure of G(D) when D does not satisfy this condition is relatively simple and easy to understand. Our result gives a unified criterion that implies all the known results on the existence of a giant component in G(D), including both the generalizations of the MolloyReed result and results on more restrictive models. Moreover, it turns out that the heuristic argument used in all the previous works on the topic, does not extend to general degree sequences.
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20200504T07:05:36Z
Joos, Felix
Perarnau Llobet, Guillem
Rautenbach, Dieter
Reed, Bruce
The traditional ErdosRenyi model of a random network is of little use in modelling the type of complex networks which modern researchers study. In this graph, every pair of vertices is equally likely to be connected by an edge. However, 21st century networks are of diverse nature and usually exhibit inhomogeneity among their nodes. This motivates the study, for a fixed degree sequence D=(d1, ..., dn), of a uniformly chosen simple graph G(D) on {1, ..., n} where the vertex i has degree di. In this paper, we study the existence of a giant component in G(D). A heuristic argument suggests that a giant component in G(D) will exist provided that the sum of the squares of the degrees is larger than twice the sum of the degrees. In 1995, Molloy and Reed essentially proved this to be the case when the degree sequence D under consideration satisfies certain technical conditions [Random Structures & Algorithms, 6:161180]. This work has attracted considerable attention, has been extended to degree sequences under weaker conditions and has been applied to random models of a wide range of complex networks such as the World Wide Web or biological systems operating at a submolecular level. Nevertheless, the technical conditions on D restrict the applicability of the result to sequences where the vertices of high degree play no important role. This is a major problem since it is observed in many realworld networks, such as scalefree networks, that vertices of high degree (the socalled hubs) are present and play a crucial role. In this paper we characterize when a uniformly random graph with a fixed degree sequence has a giant component. Our main result holds for every degree sequence of length n provided that a minor technical condition is satisfied. The typical structure of G(D) when D does not satisfy this condition is relatively simple and easy to understand. Our result gives a unified criterion that implies all the known results on the existence of a giant component in G(D), including both the generalizations of the MolloyReed result and results on more restrictive models. Moreover, it turns out that the heuristic argument used in all the previous works on the topic, does not extend to general degree sequences.