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Eric Sopena - One of the best experts on this subject based on the ideXlab platform.
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On the signed chromatic number of some classes of graphs
2020Co-Authors: Julien Bensmail, Theo Pierron, Sandip Das, Soumen Nandi, Sagnik Sen, Eric SopenaAbstract:A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph $(G, \sigma)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, \pi)$ to which $(G, \sigma)$ admits a homomorphism. Homomorphisms of signed graphs have been attracting growing attention in the last decades, especially due to their strong connections to the theories of graph coloring and graph minors. These Homomorphisms have been particularly studied through the scope of the signed chromatic number. In this work, we provide new results and bounds on the signed chromatic number of several families of signed graphs (planar graphs, triangle-free planar graphs, $K_n$-minor-free graphs, and bounded-degree graphs).
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Homomorphisms of Signed Graphs
Journal of Graph Theory, 2014Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph Homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ'] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ' if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph Homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph Homomorphisms. In this work, our focus would be on the relation of Homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.
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Homomorphisms of Signed Graphs
Journal of Graph Theory, 2014Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph Homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ'] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ' if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph Homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph Homomorphisms. In this work, our focus would be on the relation of Homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.
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Homomorphisms of signed bipartite graphs
2013Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:We study the homomorphism relation between signed graphs where the underlying graph G is bipartite. We show that this notion captures the notions of chromatic number and graph Homomorphisms. In particular we will study Hadwiger's conjecture in this setting. We show that for small values of the chromatic number there are natural strengthening of this conjecture but such extensions will not work for larger chromatic numbers.
Reza Naserasr - One of the best experts on this subject based on the ideXlab platform.
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Homomorphisms of Signed Graphs
Journal of Graph Theory, 2014Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph Homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ'] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ' if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph Homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph Homomorphisms. In this work, our focus would be on the relation of Homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.
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Homomorphisms of Signed Graphs
Journal of Graph Theory, 2014Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph Homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ'] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ' if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph Homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph Homomorphisms. In this work, our focus would be on the relation of Homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.
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Homomorphisms of signed bipartite graphs
2013Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:We study the homomorphism relation between signed graphs where the underlying graph G is bipartite. We show that this notion captures the notions of chromatic number and graph Homomorphisms. In particular we will study Hadwiger's conjecture in this setting. We show that for small values of the chromatic number there are natural strengthening of this conjecture but such extensions will not work for larger chromatic numbers.
Jinyi Cai - One of the best experts on this subject based on the ideXlab platform.
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a decidable dichotomy theorem on directed graph Homomorphisms with non negative weights
Computational Complexity, 2019Co-Authors: Jinyi Cai, Xi ChenAbstract:The complexity of graph homomorphism problems has been the subject of intense study for some years. In this paper, we prove a decidable complexity dichotomy theorem for the partition function of directed graph Homomorphisms. Our theorem applies to all non-negative weighted forms of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function ZA(G) of directed graph Homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability on the other hand is algebraic and based on properties of polynomials.
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a decidable dichotomy theorem on directed graph Homomorphisms with non negative weights
arXiv: Computational Complexity, 2010Co-Authors: Jinyi Cai, Xi ChenAbstract:The complexity of graph homomorphism problems has been the subject of intense study. It is a long standing open problem to give a (decidable) complexity dichotomy theorem for the partition function of directed graph Homomorphisms. In this paper, we prove a decidable complexity dichotomy theorem for this problem and our theorem applies to all non-negative weighted form of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function Z_A(G) of directed graph Homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability is algebraic using properties of polynomials.
Edita Rollová - One of the best experts on this subject based on the ideXlab platform.
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Homomorphisms of Signed Graphs
Journal of Graph Theory, 2014Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph Homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ'] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ' if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph Homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph Homomorphisms. In this work, our focus would be on the relation of Homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.
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Homomorphisms of Signed Graphs
Journal of Graph Theory, 2014Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:A signed graph [G,Σ] is a graph G together with an assignment of signs + and - to all the edges of G where Σ is the set of negative edges. Furthermore [G,Σ1] and [G,Σ2] are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere-zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph Homomorphisms where he showed how some well-known conjectures can be captured using this notion. A signed graph [G,Σ] is said to map to [H,Σ1] if there is an equivalent signed graph [G,Σ'] of [G,Σ] and a mapping i¾?:VGi¾?VH such that i if xy∈EG then i¾?xi¾?y∈EH and iixy∈Σ' if and only if i¾?xi¾?y∈Σ1. The chromatic number of a signed graph [G,Σ] can then be defined as the smallest order of a homomorphic image of [G,Σ]. Capturing the notion of graph homomorphism order, signed graph Homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph Homomorphisms. In this work, our focus would be on the relation of Homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.
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Homomorphisms of signed bipartite graphs
2013Co-Authors: Reza Naserasr, Edita Rollová, Eric SopenaAbstract:We study the homomorphism relation between signed graphs where the underlying graph G is bipartite. We show that this notion captures the notions of chromatic number and graph Homomorphisms. In particular we will study Hadwiger's conjecture in this setting. We show that for small values of the chromatic number there are natural strengthening of this conjecture but such extensions will not work for larger chromatic numbers.
Xi Chen - One of the best experts on this subject based on the ideXlab platform.
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a decidable dichotomy theorem on directed graph Homomorphisms with non negative weights
Computational Complexity, 2019Co-Authors: Jinyi Cai, Xi ChenAbstract:The complexity of graph homomorphism problems has been the subject of intense study for some years. In this paper, we prove a decidable complexity dichotomy theorem for the partition function of directed graph Homomorphisms. Our theorem applies to all non-negative weighted forms of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function ZA(G) of directed graph Homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability on the other hand is algebraic and based on properties of polynomials.
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a decidable dichotomy theorem on directed graph Homomorphisms with non negative weights
arXiv: Computational Complexity, 2010Co-Authors: Jinyi Cai, Xi ChenAbstract:The complexity of graph homomorphism problems has been the subject of intense study. It is a long standing open problem to give a (decidable) complexity dichotomy theorem for the partition function of directed graph Homomorphisms. In this paper, we prove a decidable complexity dichotomy theorem for this problem and our theorem applies to all non-negative weighted form of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function Z_A(G) of directed graph Homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability is algebraic using properties of polynomials.
